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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 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 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 ...
The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. [3] [4] The center of an excircle is the intersection of the internal bisector of one angle (at vertex A, for example) and the external bisectors of the other two.
A convex quadrilateral is ex-tangential if and only if there are six concurrent angles bisectors: the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.
Intersection of the symmedians – the reflection of each median about the corresponding angle bisector Note that, in general, the incenter is not the same as the centroid ; the centroid has barycentric coordinates 1 : 1 : 1 (these being proportional to actual signed areas of the triangles BGC , CGA , AGB , where G = centroid.)
In many cases, triangles can be solved given three pieces of information some of which are the lengths of the triangle's medians, altitudes, or angle bisectors. Posamentier and Lehmann [ 7 ] list the results for the question of solvability using no higher than square roots (i.e., constructibility ) for each of the 95 distinct cases; 63 of these ...
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 ...