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 here 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 these ratios.
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
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.
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?
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.
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