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The bisectors of two exterior angles and the bisector of the other interior angle are concurrent. [3]: p.149 Three intersection points, each of an external angle bisector with the opposite extended side, are collinear (fall on the same line as each other). [3]: p. 149
The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof. An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.
The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...
In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear. [ 1 ] : p. 149 In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are ...
Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons. In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point). [18]: 149 In a triangle, three intersection points, each of an external angle bisector with the opposite extended side ...
The Steiner–Lehmus theorem, a theorem in elementary geometry, was formulated by C. L. Lehmus and subsequently proved by Jakob Steiner. It states: Every triangle with two angle bisectors of equal lengths is isosceles. The theorem was first mentioned in 1840 in a letter by C. L. Lehmus to C. Sturm, in which he asked for a purely geometric proof ...
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The exterior angle theorem is not valid in spherical geometry nor in the related elliptical geometry. Consider a spherical triangle one of whose vertices is the North Pole and the other two lie on the equator. The sides of the triangle emanating from the North Pole (great circles of the sphere) both meet the equator at right angles, so this ...