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© Copyright 2003, Jim Loy
This theorem
is not found in Euclid: If two angle bisectors of a triangle are equal, then
the triangle is isosceles. In the diagram, angle bisectors BD and CE are of
equal length, show that AB=AC. This is called the Steiner-Lehmus theorem, and
it is surprisingly difficult to prove. You might want to try to prove it.
I will be showing a proof or two here. But they are under construction.
Addendum: Here is a clue about what the difficulty is. I drew
a triangle and bisected the two base angles. I kept the base and one angle
bisector (upper left to lower right) at constant lengths. As I changed the far
right angle, I traced the path of the larger red point, and got the curve
shown. Certainly this curve is of fourth degree or higher. What this means is
that if you have the length of the base and the length of one angle bisector
and the size of the angle that it bisects, then you will have great difficulty
calculating the length of the other angle bisector.
So, a straightforward attempt at a proof, using equations giving the lengths of the angle bisectors, may be very difficult to find, as the equations may be of fourth degree or higher.