Triangle

A triangle or trigon is a polygon consisting of three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent sides; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between the base and apex is the height. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated within a unique flat plane. More generally, four points in three-dimensional Euclidean space determine a solid figure called tetrahedron. In non-Euclidean geometries, three "straight" segments also determine a "triangle", for instance, a spherical triangle or hyperbolic triangle.…

A triangle or trigon is a polygon consisting of three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent sides; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between the base and apex is the height. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated within a unique flat plane. More generally, four points in three-dimensional Euclidean space determine a solid figure called tetrahedron. In non-Euclidean geometries, three "straight" segments also determine a "triangle", for instance, a spherical triangle or hyperbolic triangle. A geodesic triangle is a region of a general two-dimensional surface enclosed by three sides that are straight relative to the surface (geodesics). A triangle is a shape with three curved sides, for instance, a circular triangle with circular-arc sides. (This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted.) Triangles are classified into different types based on their angles and the lengths of their sides. Relations between angles and side lengths are a major focus of trigonometry. In particular, the sine, cosine, and tangent functions relate side lengths and angles in right triangles. \cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos(\alpha)\cos(\beta)\cos(\gamma) = 1. T = \tfrac{1}{2}bh. This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base and height . If two sides and and their included angle \gamma are known, then the altitude can be calculated using trigonometry, , so the area of the triangle is: T = \tfrac{1}{2}ab \sin \gamma. Heron's formula, named after Heron of Alexandria, is a formula for finding the area of a triangle from the lengths of its sides a , b , c . Letting s = \tfrac12(a + b + c) be the semiperimeter, T = \sqrt{s(s - a)(s - b)(s - c)}. . Because the ratios between areas of shapes in the same plane are preserved by affine transformations, the relative areas of triangles in any affine plane can be defined without reference to a notion of distance or squares. In any affine space (including Euclidean planes), every triangle with the same base and oriented area has its apex (the third vertex) on a line parallel to the base, and their common area is half of that of a parallelogram with the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid's Elements. Given affine coordinates (such as Cartesian coordinates) , , for the vertices of a triangle, its relative oriented area can be calculated using the shoelace formula, \begin{align} T &= \tfrac12 \begin{vmatrix}x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end{vmatrix} = \tfrac12 \begin{vmatrix} x_A & x_B \\ y_A & y_B \end{vmatrix} + \tfrac12 \begin{vmatrix} x_B & x_C \\ y_B & y_C \end{vmatrix} + \tfrac12 \begin{vmatrix} x_C & x_A \\ y_C & y_A \end{vmatrix} \\ &= \tfrac12(x_Ay_B - x_By_A + x_By_C - x_Cy_B + x_Cy_A - x_Ay_C), \end{align} where | \cdot | is the matrix determinant. === Possible side lengths === The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. The sum of two side lengths can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. \frac{\overline{PA} \cdot \overline{QA}}{\overline{CA} \cdot \overline{AB}} + \frac{\overline{PB} \cdot \overline{QB}}{\overline{AB} \cdot \overline{BC}} + \frac{\overline{PC} \cdot \overline{QC}}{\overline{BC} \cdot \overline{CA}} = 1. 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 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). and the three placements of its largest square. The placement on the long side of the triangle is inscribed; the other two are not. The inscribed squares tangent their vertices to the triangle's sides is the special case of inscribed square problem, although the problem asking for a square whose vertices lie on a simple closed curve. A notable example of this figure relation is the Calabi triangle in which the vertices of every three squares are tangent to all obtuse triangle's sides. 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 a 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 , h_a from the side a , and the triangle's area T are related according to q_a=\frac{2Ta}{a^2+2T} = \frac{ah_a}{a+h_a}. The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when a^2 = 2T , q = a/2 , and the altitude of the triangle from the base of length a is equal to a . The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2\sqrt{2}/3 . Both of these extreme cases occur for the isosceles right triangle. 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 convex polygon with area T can be inscribed in a triangle of area at most equal to 2T . Equality holds only if the polygon is a parallelogram. ==External links== Clark Kimberling: Encyclopedia of triangle centers. Lists some 5200 interesting points associated with any triangle.

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