The Angle Bisector of a Triangle: Properties and Theorem Explained

angle bisector of a triangle

a line segment that bisects one of the vertex angles of a triangle

The angle bisector of a triangle is a line segment that divides an angle into two congruent angles. It is a line that starts from the vertex of the angle and goes through the opposite side, dividing the angle into two equal parts.

In a triangle ABC, if the angle bisector of angle A meets the opposite side BC at point D, then:

1. AD is the angle bisector of angle A.
2. BD/DC = AB/AC (This is called the Angle Bisector Theorem).

The angle bisector is a significant line in a triangle because it has some useful properties, some of which are:

1. The angle bisectors of all three angles of a triangle meet at a point called the incenter of the triangle.
2. The incenter is equidistant from all three sides of the triangle. This point is the center of the incircle which is the circle that is tangent to all three sides of the triangle.
3. The angle bisectors divide the triangle into three smaller triangles that have equal ratios of the lengths of sides to the semiperimeter of the triangle.

In summary, the angle bisector of a triangle is a line that divides an angle into two congruent parts, and it has valuable geometric properties that are used in various geometrical problems.

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