Department of Mathematics Education

Dr. J. Wilson, EMAT 6690

__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 above we have

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

**Click here** to investigate
a GSP Sketch to see if these ratios remain equivalent for all
types of 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

**Click here** to view proof

__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

**Click here** to investigate
a GSP Sketch to see if these ratios remain equivalent for all
types of triangles.

**Click here** to view proof

If you have any comments concerning this investigation that
would be useful, especially for use at the high school level,
please send e-mail to **esiwdivad@yahoo.com.**

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