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For the exterior angle bisectors in a non-equilateral triangle there exist similar equations for the ratios of the lengths of triangle sides. More precisely if the exterior angle bisector in A intersects the extended side BC in E, the exterior angle bisector in B intersects the extended side AC in D and the exterior angle bisector in C ...
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 ...
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 divides an angle of less than 180° into two equal angles.
The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line segments). [3] Only one of these angles contains the third side of the triangle in its interior, and this angle is called an interior angle of the triangle. [ 4 ]
Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter. Other sets of lines associated with a triangle are concurrent as well. For example: Any median (which is necessarily a bisector of the triangle's area) is concurrent with two ...
The extended base of a triangle (a particular case of an extended side) is the line that contains the base. When the triangle is obtuse and the base is chosen to be one of the sides adjacent to the obtuse angle , then the altitude dropped perpendicularly from the apex to the base intersects the extended base outside 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. Its radius is called the inradius.
When one of I or J is the incenter, this is the trillium theorem, with line IJ as the (internal) angle bisector of one of the triangle's angles. However, it is also true when I and J are both excenters; in this case, line IJ is the external angle bisector of one of the triangle's angles. [14]