**Exploration**

Construct any triangle. Construct an angle bisector in the triangle and draw the segment along the angle bisector from the vertex to the intersection with the opposite side.

Measure the **ratio** of the adjacent sides. In the triangle
pictured we have

Measure the **ratio** of the segments cut off by the bisector
on the opposite side. In the triangle pictured here we have

Repeat for many triangles. Click here for a GSP Sketch to explore some triangles.

**To Prove**

The bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides.

That is, for any triangle ABC, the bisector of the angle at C divides the opposite side into segments of length x and y such that

**EXTENSION**

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.

That is, the external bisector of the angle at C externally divides the side AB at D such that