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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.

 

Click here for 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 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.