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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:
Bisect one of the angles made by these two lines and name the angle bisector b. Using a hyperbolic ruler, construct a line c such that c is perpendicular to b and parallel to a. As a result, c is also parallel to a', making c the common parallel to lines a and a'. [3] Case 2: a and a' are parallel to each other
To bisect an angle with straightedge and compass, one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector.
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.
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]
Draw the incenter by intersecting angle bisectors. Draw a line through I {\displaystyle I} perpendicular to the line A I {\displaystyle AI} , touching lines A B {\displaystyle AB} and A C {\displaystyle AC} at points D {\displaystyle D} and E {\displaystyle E} respectively.
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.
In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. [1] [2] Medians and angle bisectors are special cases of cevians.