## 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.

Click

herefor a GSP Sketch to explore these ratios.

## 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

exteriorangle of a triangle divides the opposite sideexternallyinto 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

notequal to 1? Proof?

Build a GSP sketch to draw this locus. Clickhereto see a GSP animation.

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