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Consider a triangle ABC.Let the angle bisector of angle ∠ A intersect side BC at a point D between B and C.The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of the length of side AB to the length of side AC:
If the internal bisector of angle A in triangle ABC has length and if this bisector divides the side opposite A into segments of lengths m and n, then [3]: p.70 + = where b and c are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportion b:c.
Construct the angle bisector of and extend it to meet BC at X. AB = AC and AX is equal to itself. Furthermore, =, so, applying side-angle-side, triangle BAX and triangle CAX are congruent. It follows that the angles at B and C are equal. Legendre uses a similar construction in Éléments de géométrie, but taking X to be the midpoint of BC. [16]
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.
The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry". [1]
Some modern treatments (not Euclid's) of the proof of the theorem that the base angles of an isosceles triangle are congruent start like this: Let ABC be a triangle with side AB congruent to side AC. Draw the angle bisector of angle A and let D be the point at which it meets side BC. And so on.
Anticomplementary triangle; Orthic triangle; The triangle whose vertices are the points of contact of the incircle with the sides of ABC; Tangential triangle; The triangle whose vertices are the points of contacts of the excircles with the respective sides of triangle ABC; The triangle formed by the bisectors of the external angles of triangle ABC
The -excircle of triangle is unique. Let be a transformation defined by the composition of an inversion centered at with radius and a reflection with respect to the angle bisector on .