Bisector of an Angle of a Triangle

The following is an exploration with proofs involving any triangle and an angle bisector in the triangle.

We will first look at any given triangle ABC. An angle bisector is constructed at vertex C and meets the opposite side at point D. We will refer to segment BC as a, segment CA as b, segment BD as x, and segment DA as y.

Let's explore the relationship of he ratios of side b to a and of y to x. Click here to go to a GSP exploration.

Notice that the ratio of the adjacent sides of the triangle and the ratio of the segments cut off by the bisector are the same. Let's look at the proof for this. Refer to the figure below for the proof.

are corresponding angles on parallel lines.

Next we will go on to prove that the bisector of an exterior angle of a triangle divides the opposite side externally into segments that are proportional to the adjacent sides.