Bisector of An Angle of a triangle.

### 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.

### 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

Hint

### 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 M such that

Hint: Draw AE parallel to CD.

### Extension

Given a set of triangles all having the same base AB. What is the locus of the vertex C in the ratio of the sides adjacent to C is 1? Proof?

### Extension

Given a set of triangles all having the same base AB. What is the locus of the vertex C in the ratio of the sides adjacent to C is not equal to 1? Proof?

Build a GSP sketch to draw this locus. Click here to see a GSP animation.