A line consisting of points equidistant from the sides of a triangle is called the bisector.
This line is called inner bisector for inner angles and outer bisector for outer angles.
For a bisector, the length of the line segment between the corner and the point where the opposite edge is intersected is called the bisector length.
Inner bisector properties
Inner bisector theorem: In a triangle, the bisector length divides the opposite side in proportion to its neighboring sides.
In a triangle, inner bisectors intersect at a fixed point. This intersection point is called the center of the triangle’s inner tangent circle. The radius of the inner tangent circle is denoted by r.
From any point taken on the bisector of an angle, the perpendiculars descending to the arms of the angle are equal to each other. Also, the lengths of the sides separated by the perpendiculars from the corner are equal to each other.
In a triangle, the bisector length of an angle divides the area of the triangle in proportion to its neighboring sides.
In a triangle, the areas of the triangles that we obtain by joining the intersection point of the inner bisectors to the corners are proportional to the length of the sides.
The square of the inner bisector length in a triangle is equal to the difference in the product of the side lengths separated by the product of adjacent sides with respect to the bisector.
Outer bisector properties
Outer bisector theorem: In a triangle, the angle between the inner bisector of the same corner and the outer bisector is 90 degrees, and the ratio between the parts separated by the inner bisector and the outer bisector is equal to the distance of the outer bisector to the other corners.