To show that the angle bisector of an angle in a triangle splits the opposite side in the ratio of the two adjacent sides.
My first proof used angles in the same segment. See
https://howardat58.wordpress.com/2014/05/14/interesting-geometry-result/
Several tries later (today) and i came up with this annoyingly simple proof:
Saving school math 
Both are nice proofs!
Reblogged this on Mathpresso.
This is the same proof offered in Jurgensen, Brown, and Jurgensen’s Geometry from the late 80s/early 90s. We were just talking about it in class a week ago!
I love this proof. It took me a while to get there, and then I thought “Unfair ! This is too simple.”.
If you draw the perpendicular to the bisector through the top point of the triangle you get another point on the base, outside the triangle. The result is also true for that point. This can lead to cross ratios and harmonic ranges in projective geometry. ……….
Ooh, yes, that’s quite nicely done.