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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 'exterior' or 'external bisector' is the line that divides the supplementary angle (of 180° minus the original angle), formed by one side forming the original angle and the extension of ...
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.
The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. [20] Thales' theorem implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. [ 21 ]
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.
A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a radial elliptic trajectory.
Definition. An ellipse that is tangent to the sides of a triangle ... The major axis of the triangle's Steiner inellipse is the inner bisector of ...
The square of each internal angle bisector of an integer triangle is rational, because the general triangle formula for the internal angle bisector of angle A is () / (+) where s is the semiperimeter (and likewise for the other angles' bisectors).
Draw the lines AX, BX and CX and their reflections in the internal bisectors of the angles at the vertices A, B, C respectively. The reflected lines are concurrent and the point of concurrence is the isogonal conjugate Y of X. Let the cevians AY, BY, CY meet the opposite sidelines of triangle ABC at A' , B' , C' respectively.