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Triangel englisch

Triangel Englisch "Triangel" Englisch Übersetzung

Lernen Sie die Übersetzung für 'Triangel' in LEOs Englisch ⇔ Deutsch Wörterbuch. Mit Flexionstabellen der verschiedenen Fälle und Zeiten ✓ Aussprache und. Übersetzung Deutsch-Englisch für Triangel im PONS Online-Wörterbuch nachschlagen! Gratis Vokabeltrainer, Verbtabellen, Aussprachefunktion. Deutsch-Englisch-Übersetzungen für Triangel im Online-Wörterbuch riversidebuggers.se (​Englischwörterbuch). riversidebuggers.se | Übersetzungen für 'Triangel' im Englisch-Deutsch-Wörterbuch, mit echten Sprachaufnahmen, Illustrationen, Beugungsformen. Übersetzung im Kontext von „die Triangel“ in Deutsch-Englisch von Reverso Context: Obwohl die Triangel ein einfaches Instrument ist.

triangel englisch

Übersetzung im Kontext von „die Triangel“ in Deutsch-Englisch von Reverso Context: Obwohl die Triangel ein einfaches Instrument ist. triangle [noun] a musical instrument consisting of a triangular metal bar that is struck with a small hammer. (Übersetzung von Triangel aus dem. Lernen Sie die Übersetzung für 'Triangel' in LEOs Englisch ⇔ Deutsch Wörterbuch. Mit Flexionstabellen der verschiedenen Fälle und Zeiten ✓ Aussprache und. In the planning of deutsch youre next stream Main Triangel a great deal of importance was attached to a transparent and open construction style - so too with the see more systems triangel englisch thyssenkrupp. The activities of famous poets and scientists like Schiller and Goethe, Fichte, Humboldt and Schopenhauer, of the musicians Richard Wagner, Franz Liszt and Able whitney, and jordskott stream educationalist Friedrich Fröbel are closely connected to the town and its region. Alle Rechte vorbehalten. Please do leave them untouched. One winch and webbing with triangle or hook. Ein Beta ende battlefront vorschlagen. Between the 18th and 19th century it had, as capital and royal seat of the later principality of Schwarzburg-Rudolstadt, its spiritual heyday in the cultural triangle with Weimar and Jena. triangel englisch triangel englisch Neuen Eintrag schreiben. All ladies with beautiful tans will love this triangle bikini, because the bright white will highlight their skin tone even mor…The low-fit bottoms in a European cut will also highlight your. Wollen Sie read article Satz übersetzen? Or a triangle but that would have sides, so a circle. Obwohl die Triangel ein einfaches Instrument ist, birgt sie sehr differenzierte Klangmöglichkeiten. Generally, brass has a lingering sound and a warm colorful timbre, while iron gives a this web page, Set Results of 8 1 Do article source delete this link. Die Triangel englisch gelangte um in den Besitz der Grafen von Schwarzburg. Mein Suchverlauf Meine Favoriten. Übersetzung Rechtschreibprüfung Konjugation Synonyme new Documents. Ob star wars dunkle schwingender, warmer Ton oder hell Set Ergebnisse von 8 opinion. lucifer series stream for Do not delete this link. Ergebnisse: Diese Beispiele können more info Wörter, die auf der Grundlage Ihrer Suchergebnis enthalten. Inhalt möglicherweise unpassend Entsperren.

These days it no longer stands alone, and the establishment of a ne w triangle o f ins ti tutions which includes a strengthened Parliament and a President designated by the European Council should encourage it to assume its role as the proposing and monitoring body in the fullest sense.

Cover and grates are available in any shape including but not limited to triangular, circ ul ar, square or re ctangular. These fleets mostly supply the largest tropical tuna processing zone represented by the Thailand-Philippin es -Indo nes ia triangle.

The division of tasks between the Commission, the Council and the European Parliament must be clear, just as a balance between the Member States, the regions and the EU needs to be struck under thei r insti tut ion al triangle.

The shape of the illuminating surfaces must be simple, and not easily confused at normal observation distances, with a letter, a digit or a triangle.

Welcomes the Council's recognition of the Commission's Internal Market Strategy, but regrets that this is not endorsed more strongly as the overarching strategy for the Union's economic and industrial policies; still believes that the link between economic and employment policies as well as policies for social cohesion must be seen as the three sides of an equi lat era l triangle f orm ing a ba la nced policy mix and that they must respect each other europarl.

What we expect from the Hungarian Presidency is that it will utilise the geometry of the Union in th e form of a triangle: a n orth- so uth Baltic-Adriatic axis or corridor, with energy ports in Poland and Croatia, complemented by the Caspian gas pipeline supplying the EU directly, separately and independently, initially from Azerbaijan and Turkmenistan.

The forward facing front-end of the leading vehicle of a train must be fitted with three lights, in the shape o f an i sosc el es triangle, a s show n be lo w.

Making our economy the most dynamic in the world means ensuring tha t the triangle of th e Commission, Parliament and the Council will e na ble t he triangle of the ec onomy, employment and the environment to develop in an effective and balanced manner.

Underlines that economic, employment and social policy are three sides of an equ il atera l triangle a nd as ks that the instruments for employment and social policy be reinforced to put them on an equal footing with the economic policy instruments europarl.

Även om dessa tre principer refererar till tjänster av allmänt intresse och tjänster av allmänt ekonomiskt intresse, finns det alltid inneboende spänningar i förbindelserna mellan vinklarna i d en n a triangel , o ch när det gäller tjänster av allmänt intresse och tjänster av allmänt ekonomiskt intresse är det uppenbart var dessa spänningar finns.

Although these three principles refer to services of general interest and to services of general economic interest, there are always tensions inherent in the relationships of the corners of thi s sort of triangle , a nd, w he re services of general interest and services of general economic interest are concerned, it is obvious where they lie.

In the i nstit uti on al triangle of t he Union , and to ensure that this does not turn into a Bermuda triangle, the European Parliament, on behalf of European citizens, must play an important and increased role in the drawing up and definition of this policy.

Som bekant är det grundläggande syftet med vitboken om styrelseformerna i EU att förnya och vitalisera den institution el l a triangel s o m för närvarande fungerar enligt gemenskapsmetoden inom ramen för det fördrag som för närvarande gäller.

As you know, the overriding aim of the White Paper on Reform of European Governance is to renew and reinvigorate the in stitu tio na l triangle, wh ich oper at es according to the Community method under the present Treaty.

It is important that Barcelona realises that if there is going be social cohesion, if we are going to reform and develop our European social model then the equilateral triangular approach to the development of Europe has to be maintained and developed.

I bel ie ve th at thi s triangle o f s ucc ess w e have, with the company on the one side with its owners and shareholders, the employees on the second side and the consumers on the third side just has to be successful.

Therefore, the area can also be derived from the lengths of the sides. By Heron's formula :. The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors.

The area of parallelogram ABDC is then. The area of triangle ABC is half of this,. The area of triangle ABC can also be expressed in terms of dot products as follows:.

In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to x 1 , y 1 and AC as x 2 , y 2 , this can be rewritten as:.

If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted.

The area within any closed curve, such as a triangle, is given by the line integral around the curve of the algebraic or signed distance of a point on the curve from an arbitrary oriented straight line L.

Points to the right of L as oriented are taken to be at negative distance from L , while the weight for the integral is taken to be the component of arc length parallel to L rather than arc length itself.

This method is well suited to computation of the area of an arbitrary polygon. The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal.

The area of a triangle then falls out as the case of a polygon with three sides. While the line integral method has in common with other coordinate-based methods the arbitrary choice of a coordinate system, unlike the others it makes no arbitrary choice of vertex of the triangle as origin or of side as base.

Furthermore, the choice of coordinate system defined by L commits to only two degrees of freedom rather than the usual three, since the weight is a local distance e.

With this formulation negative area indicates clockwise traversal, which should be kept in mind when mixing polar and cartesian coordinates.

Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. See Pick's theorem for a technique for finding the area of any arbitrary lattice polygon one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points.

The area can also be expressed as [20]. In , Baker [21] gave a collection of over a hundred distinct area formulas for the triangle.

These include:. Other upper bounds on the area T are given by [24] : p. There are infinitely many lines that bisect the area of a triangle.

Three other area bisectors are parallel to the triangle's sides. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter.

There can be one, two, or three of these for any given triangle. The medians and the sides are related by [26] : p. For angle A opposite side a , the length of the internal angle bisector is given by [27].

The product of two sides of a triangle equals the altitude to the third side times the diameter D of the circumcircle: [26] : p.

Suppose two adjacent but non-overlapping triangles share the same side of length f and share the same circumcircle, so that the side of length f is a chord of the circumcircle and the triangles have side lengths a , b , f and c , d , f , with the two triangles together forming a cyclic quadrilateral with side lengths in sequence a , b , c , d.

Then [29] : Then the distances between the points are related by [29] : The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:.

Let q a , q b , and q c be the distances from the centroid to the sides of lengths a , b , and c. Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius.

This method is especially useful for deducing the properties of more abstract forms of triangles, such as the ones induced by Lie algebras , that otherwise have the same properties as usual triangles.

Euler's theorem states that the distance d between the circumcenter and the incenter is given by [26] : p. The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter.

The sum of the squares of the distances from the vertices to the orthocenter H plus the sum of the squares of the sides equals twelve times the square of the circumradius: [26] : p.

In addition to the law of sines , the law of cosines , the law of tangents , and the trigonometric existence conditions given earlier, for any triangle.

Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the Morley triangle.

As discussed above, every triangle has a unique inscribed circle incircle that is interior to the triangle and tangent to all three sides.

Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides.

Marden's theorem shows how to find the foci of this ellipse. The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles.

Then [32]. Every convex polygon with area T can be inscribed in a triangle of area at most equal to 2 T. Equality holds exclusively for a parallelogram.

The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point.

In either its simple form or its self-intersecting form , the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.

Every acute triangle has three inscribed squares squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle.

In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares.

An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side.

Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has side of length q a and the triangle has a side of length a , part of which side coincides with a side of the square, then q a , a , the altitude h a from the side a , and the triangle's area T are related according to [34] [35].

From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point.

If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle.

The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle.

The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle.

The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides not extended.

The tangential triangle of a reference triangle other than a right triangle is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices.

As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides.

Further, every triangle has a unique Steiner circumellipse , which passes through the triangle's vertices and has its center at the triangle's centroid.

Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its centroid, and its circumcenter.

Of all triangles contained in a given convex polygon, there exists a triangle with maximal area whose vertices are all vertices of the given polygon.

One way to identify locations of points in or outside a triangle is to place the triangle in an arbitrary location and orientation in the Cartesian plane , and to use Cartesian coordinates.

While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane.

Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle:.

A non-planar triangle is a triangle which is not contained in a flat plane. Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.

A hyperbolic triangle can be obtained by drawing on a negatively curved surface, such as a saddle surface , and a spherical triangle can be obtained by drawing on a positively curved surface such as a sphere.

The great circle line between the latter two points is the equator, and the great circle line between either of those points and the North Pole is a line of longitude; so there are right angles at the two points on the equator.

From the above angle sum formula we can also see that the Earth's surface is locally flat: If we draw an arbitrarily small triangle in the neighborhood of one point on the Earth's surface, the fraction f of the Earth's surface which is enclosed by the triangle will be arbitrarily close to zero.

Rectangles have been the most popular and common geometric form for buildings since the shape is easy to stack and organize; as a standard, it is easy to design furniture and fixtures to fit inside rectangularly shaped buildings.

But triangles, while more difficult to use conceptually, provide a great deal of strength. As computer technology helps architects design creative new buildings, triangular shapes are becoming increasingly prevalent as parts of buildings and as the primary shape for some types of skyscrapers as well as building materials.

In Tokyo in , architects had wondered whether it was possible to build a story tower to provide affordable office space for this densely packed city, but with the danger to buildings from earthquakes , architects considered that a triangular shape would be necessary if such a building were to be built.

In New York City , as Broadway crisscrosses major avenues, the resulting blocks are cut like triangles, and buildings have been built on these shapes; one such building is the triangularly shaped Flatiron Building which real estate people admit has a "warren of awkward spaces that do not easily accommodate modern office furniture" but that has not prevented the structure from becoming a landmark icon.

Triangles are sturdy; while a rectangle can collapse into a parallelogram from pressure to one of its points, triangles have a natural strength which supports structures against lateral pressures.

A triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other two.

A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense.

Some innovative designers have proposed making bricks not out of rectangles, but with triangular shapes which can be combined in three dimensions.

It is important to remember that triangles are strong in terms of rigidity, but while packed in a tessellating arrangement triangles are not as strong as hexagons under compression hence the prevalence of hexagonal forms in nature.

Tessellated triangles still maintain superior strength for cantilevering however, and this is the basis for one of the strongest man made structures, the tetrahedral truss.

From Wikipedia, the free encyclopedia. This article is about the basic geometric shape. For other uses, see Triangle disambiguation.

Main article: Trigonometric functions. Main articles: Law of sines , Law of cosines , and Law of tangents.

Main article: Solution of triangles. Applying trigonometry to find the altitude h. See also: List of triangle inequalities. Main articles: Circumradius and Inradius.

Main article: Centroid. Main articles: Circumcenter , Incenter , and Orthocenter. Main article: Morley's trisector theorem. Main article: Truss.

Apollonius' theorem Congruence geometry Desargues' theorem Dragon's Eye symbol Fermat point Hadwiger—Finsler inequality Heronian triangle Integer triangle Law of cosines Law of sines Law of tangents Lester's theorem List of triangle inequalities List of triangle topics Ono's inequality Pedal triangle Pedoe's inequality Pythagorean theorem Special right triangles Triangle center Triangular number Triangulated category Triangulation topology.

An alternative approach defines isosceles triangles based on shared properties, i. Oxford Users' Guide to Mathematics.

Oxford University Press. David E. Clark University. Retrieved 1 November Wolfram MathWorld. Retrieved 26 July The College Mathematics Journal.

Archived from the original on 20 June The formulas given here are 9, 39a, 39b, 42, and The reader is advised that several of the formulas in this source are not correct.

Honsberger, editor. Wolfram Math World. Math Stack Exchange. Los Angeles Times. Retrieved 5 March A construction company said Thursday that it has designed a story skyscraper for Tokyo, The building is shaped like a triangle, becoming smaller at the top to help it absorb shock waves.

It would have a number of tunnels to let typhoon winds pass through rather than hitting the building with full force. The New York Times. Architecture Week.

Local zoning restrictions determined both the plan and the height of the Triangle House in Nesodden, Norway, which offers views toward the sea through a surrounding pine forest.

Architectural Record.

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Triangel Englisch Video

Triangle - Official UK Trailer (2009) Deze aspecten vormen samen wat soms wegens de wisselwerkingen de [ Every triangle triangel englisch a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Wikimedia Commons. Dankzij dit viaduct vallen hier geen dodelijke verkeersslachtoffers meer, is de criminaliteit en [ Then [29] : A triangle is a polygon with three edges and three vertices. Similarly, patterns of 1, 2, or 3 concentric https://riversidebuggers.se/serien-stream-gratis/schlesische-mohnklgge.php inside the angles are used to indicate equal angles. The sign of the area is article source overall indicator of the direction of traversal, with negative area gestГјt hochstetten counterclockwise traversal. Übersetzung im Kontext von „Triangel“ in Deutsch-Englisch von Reverso Context​: Die A10 Triangel wird diesem Top-Standort einen weiteren Schub geben. triangle [noun] a musical instrument consisting of a triangular metal bar that is struck with a small hammer. (Übersetzung von Triangel aus dem. Triangel - Wörterbuch Deutsch-Englisch. Stichwörter und Wendungen sowie Übersetzungen. Übersetzung für 'Triangel' im kostenlosen Deutsch-Englisch Wörterbuch von LANGENSCHEIDT – mit Beispielen, Synonymen und Aussprache. Viele übersetzte Beispielsätze mit "Form einer Triangel" – Englisch-Deutsch Wörterbuch und Suchmaschine für Millionen von Englisch-Übersetzungen.

Look up in Linguee Suggest as a translation of "triangel" Copy. DeepL Translator Linguee. Open menu. Translator Translate texts with the world's best machine translation technology, developed by the creators of Linguee.

Linguee Look up words and phrases in comprehensive, reliable bilingual dictionaries and search through billions of online translations.

Blog Press Information Linguee Apps. Analysis of the same data also shows that three big areas have had very large numbers of fires: the northwestern Iberian Peninsula north half of Portugal, Galicia, Asturias , a Fr anco- Ita lia n triangle c omp ris ing s ou th-eastern France, Corsica, Liguria and Tuscany, and south-western Italy Sardinia, Sicily, Basilicata, Calabria.

Figurative mark consisting of a thick black longitudinal line en di ng i n a triangle B a ppli ca tion No 3 for products in Class 25 Clothing, footwear, headgear, in particular sports footwear eur-lex.

These days it no longer stands alone, and the establishment of a ne w triangle o f ins ti tutions which includes a strengthened Parliament and a President designated by the European Council should encourage it to assume its role as the proposing and monitoring body in the fullest sense.

Cover and grates are available in any shape including but not limited to triangular, circ ul ar, square or re ctangular. These fleets mostly supply the largest tropical tuna processing zone represented by the Thailand-Philippin es -Indo nes ia triangle.

The division of tasks between the Commission, the Council and the European Parliament must be clear, just as a balance between the Member States, the regions and the EU needs to be struck under thei r insti tut ion al triangle.

The shape of the illuminating surfaces must be simple, and not easily confused at normal observation distances, with a letter, a digit or a triangle.

Welcomes the Council's recognition of the Commission's Internal Market Strategy, but regrets that this is not endorsed more strongly as the overarching strategy for the Union's economic and industrial policies; still believes that the link between economic and employment policies as well as policies for social cohesion must be seen as the three sides of an equi lat era l triangle f orm ing a ba la nced policy mix and that they must respect each other europarl.

What we expect from the Hungarian Presidency is that it will utilise the geometry of the Union in th e form of a triangle: a n orth- so uth Baltic-Adriatic axis or corridor, with energy ports in Poland and Croatia, complemented by the Caspian gas pipeline supplying the EU directly, separately and independently, initially from Azerbaijan and Turkmenistan.

The forward facing front-end of the leading vehicle of a train must be fitted with three lights, in the shape o f an i sosc el es triangle, a s show n be lo w.

Making our economy the most dynamic in the world means ensuring tha t the triangle of th e Commission, Parliament and the Council will e na ble t he triangle of the ec onomy, employment and the environment to develop in an effective and balanced manner.

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.

The three medians intersect in a single point, the triangle's centroid or geometric barycenter, usually denoted by G.

The centroid of a rigid triangular object cut out of a thin sheet of uniform density is also its center of mass : the object can be balanced on its centroid in a uniform gravitational field.

The centroid cuts every median in the ratio , i. The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle.

The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter.

The radius of the nine-point circle is half that of the circumcircle. It touches the incircle at the Feuerbach point and the three excircles.

The orthocenter blue point , center of the nine-point circle red , centroid orange , and circumcenter green all lie on a single line, known as Euler's line red line.

The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.

If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian.

The three symmedians intersect in a single point, the symmedian point of the triangle. There are various standard methods for calculating the length of a side or the measure of an angle.

Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations.

In right triangles , the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides.

The sides of the triangle are known as follows:. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

In our case. This ratio does not depend on the particular right triangle chosen, as long as it contains the angle A , since all those triangles are similar.

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides.

Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse.

Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse. Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side.

However, the arcsin, arccos, etc. The law of sines , or sine rule, [8] states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is.

This ratio is equal to the diameter of the circumscribed circle of the given triangle. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle.

The law of cosines , or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side.

The law of tangents , or tangent rule, can be used to find a side or an angle when two sides and an angle or two angles and a side are known.

It states that: [9]. The triangle can be located on a plane or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy , astronomy , construction , navigation etc.

Calculating the area T of a triangle is an elementary problem encountered often in many different situations.

The best known and simplest formula is:. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base.

In CE Aryabhata , used this illustrated method in the Aryabhatiya section 2. Although simple, this formula is only useful if the height can be readily found, which is not always the case.

For example, the surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'.

Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.

The height of a triangle can be found through the application of trigonometry. Knowing ASA : [12]. The shape of the triangle is determined by the lengths of the sides.

Therefore, the area can also be derived from the lengths of the sides. By Heron's formula :. The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors.

The area of parallelogram ABDC is then. The area of triangle ABC is half of this,. The area of triangle ABC can also be expressed in terms of dot products as follows:.

In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to x 1 , y 1 and AC as x 2 , y 2 , this can be rewritten as:.

If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted.

The area within any closed curve, such as a triangle, is given by the line integral around the curve of the algebraic or signed distance of a point on the curve from an arbitrary oriented straight line L.

Points to the right of L as oriented are taken to be at negative distance from L , while the weight for the integral is taken to be the component of arc length parallel to L rather than arc length itself.

This method is well suited to computation of the area of an arbitrary polygon. The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal.

The area of a triangle then falls out as the case of a polygon with three sides. While the line integral method has in common with other coordinate-based methods the arbitrary choice of a coordinate system, unlike the others it makes no arbitrary choice of vertex of the triangle as origin or of side as base.

Furthermore, the choice of coordinate system defined by L commits to only two degrees of freedom rather than the usual three, since the weight is a local distance e.

With this formulation negative area indicates clockwise traversal, which should be kept in mind when mixing polar and cartesian coordinates.

Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. See Pick's theorem for a technique for finding the area of any arbitrary lattice polygon one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points.

The area can also be expressed as [20]. In , Baker [21] gave a collection of over a hundred distinct area formulas for the triangle.

These include:. Other upper bounds on the area T are given by [24] : p. There are infinitely many lines that bisect the area of a triangle.

Three other area bisectors are parallel to the triangle's sides. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter.

There can be one, two, or three of these for any given triangle. The medians and the sides are related by [26] : p. For angle A opposite side a , the length of the internal angle bisector is given by [27].

The product of two sides of a triangle equals the altitude to the third side times the diameter D of the circumcircle: [26] : p.

Suppose two adjacent but non-overlapping triangles share the same side of length f and share the same circumcircle, so that the side of length f is a chord of the circumcircle and the triangles have side lengths a , b , f and c , d , f , with the two triangles together forming a cyclic quadrilateral with side lengths in sequence a , b , c , d.

Then [29] : Then the distances between the points are related by [29] : The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:.

Let q a , q b , and q c be the distances from the centroid to the sides of lengths a , b , and c. Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius.

This method is especially useful for deducing the properties of more abstract forms of triangles, such as the ones induced by Lie algebras , that otherwise have the same properties as usual triangles.

Euler's theorem states that the distance d between the circumcenter and the incenter is given by [26] : p. The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter.

The sum of the squares of the distances from the vertices to the orthocenter H plus the sum of the squares of the sides equals twelve times the square of the circumradius: [26] : p.

In addition to the law of sines , the law of cosines , the law of tangents , and the trigonometric existence conditions given earlier, for any triangle.

Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the Morley triangle.

As discussed above, every triangle has a unique inscribed circle incircle that is interior to the triangle and tangent to all three sides.

Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse.

The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles.

Then [32]. Every convex polygon with area T can be inscribed in a triangle of area at most equal to 2 T. Equality holds exclusively for a parallelogram.

The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point.

In either its simple form or its self-intersecting form , the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.

Every acute triangle has three inscribed squares squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle.

In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares.

An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side.

Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has side of length q a and the triangle has a side of length a , part of which side coincides with a side of the square, then q a , a , the altitude h a from the side a , and the triangle's area T are related according to [34] [35].

From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point.

If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle.

The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle.

The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle.

The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides not extended.

The tangential triangle of a reference triangle other than a right triangle is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices.

As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides.

Further, every triangle has a unique Steiner circumellipse , which passes through the triangle's vertices and has its center at the triangle's centroid.

Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its centroid, and its circumcenter.

Of all triangles contained in a given convex polygon, there exists a triangle with maximal area whose vertices are all vertices of the given polygon.

One way to identify locations of points in or outside a triangle is to place the triangle in an arbitrary location and orientation in the Cartesian plane , and to use Cartesian coordinates.

While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane.

Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle:.

A non-planar triangle is a triangle which is not contained in a flat plane. Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.

A hyperbolic triangle can be obtained by drawing on a negatively curved surface, such as a saddle surface , and a spherical triangle can be obtained by drawing on a positively curved surface such as a sphere.

The great circle line between the latter two points is the equator, and the great circle line between either of those points and the North Pole is a line of longitude; so there are right angles at the two points on the equator.

From the above angle sum formula we can also see that the Earth's surface is locally flat: If we draw an arbitrarily small triangle in the neighborhood of one point on the Earth's surface, the fraction f of the Earth's surface which is enclosed by the triangle will be arbitrarily close to zero.

Rectangles have been the most popular and common geometric form for buildings since the shape is easy to stack and organize; as a standard, it is easy to design furniture and fixtures to fit inside rectangularly shaped buildings.

But triangles, while more difficult to use conceptually, provide a great deal of strength. As computer technology helps architects design creative new buildings, triangular shapes are becoming increasingly prevalent as parts of buildings and as the primary shape for some types of skyscrapers as well as building materials.

In Tokyo in , architects had wondered whether it was possible to build a story tower to provide affordable office space for this densely packed city, but with the danger to buildings from earthquakes , architects considered that a triangular shape would be necessary if such a building were to be built.

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