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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 interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram. The bisectors of two exterior angles and the bisector of the other interior angle are concurrent. [3]: p.149
A triangle with medians (black), angle bisectors (dotted) and symmedians (red). The symmedians intersect in the symmedian point (denoted by L in the figure), the angle bisectors in the incenter I and the medians in the centroid G.
In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors: A triangle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter. Angle bisectors are rays running from each vertex of the triangle ...
An angle bisector of a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter, which is the center of the triangle's incircle. The incircle is the circle that lies inside the triangle and touches all three sides.
The point of intersection of angle bisectors of the 3 angles of triangle ABC is the incenter (denoted by I). The incircle (whose center is I) touches each side of the triangle. In geometry , the incenter of a triangle is a triangle center , a point defined for any triangle in a way that is independent of the triangle's placement or scale.
External angle bisectors (forming the excentral triangle) An excircle or escribed circle [ 2 ] of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to the extensions of the other two .
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.