Page 1: Fundamental Properties of Angle Bisectors

This page introduces the concept of üçgende açıortay (angle bisector in triangles) and outlines its key properties and relationships within a triangle.

Definition: An angle bisector in a triangle is a line segment that divides an angle into two equal parts, extending from the vertex to the opposite side.

The page covers several important properties of angle bisectors:

1. The intersection point of the three internal angle bisectors is the center of the inscribed circle (incenter) of the triangle.

2. In triangle ABC, if AD is an angle bisector and DE is parallel to BC, then |BD| = |CE|.

3. The angle bisector theorem states that the ratio of the lengths of the segments created by an angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides.

Highlight: The area of a triangle can be calculated using the semiperimeter and inradius: A(ABC) = r * s, where r is the inradius and s is the semiperimeter.

4. The sum of the lengths of two external angle bisectors and one internal angle bisector meeting at a vertex is equal to the perimeter of the triangle.

5. The relationship between angle bisectors, medians, and altitudes from the same vertex: ha < ma < va (where ha is the altitude, ma is the median, and va is the angle bisector).

Example: In a right-angled triangle, the angle bisector of the right angle is equal to half the sum of the other two sides.

The page also introduces formulas for calculating the length of internal and external angle bisectors, which are crucial for solving more complex geometric problems.