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An angle bisector divides the angle into two angles with equal measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The 'interior' or 'internal bisector' of an angle is the line, half-line, or line segment that
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.
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.
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. Sturm passed the request on to other mathematicians and Steiner was among the first to provide a solution.
In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear. [18]: 149 In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.
Area bisector of a triangle; Angle bisector of a triangle; Angle bisector theorem; Apollonius point; Apollonius' theorem; Automedian triangle; Barrow's inequality; Barycentric coordinates (mathematics) Bernoulli's quadrisection problem; Brocard circle; Brocard points; Brocard triangle; Carnot's theorem (conics) Carnot's theorem (inradius ...
The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral [24]: p.127 (that is, the four intersection points of adjacent angle bisectors are concyclic) or they are concurrent. In the latter case the quadrilateral is a tangential quadrilateral.
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
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