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The Mirror Property of Altitudes: What is this about?
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The applet may suggest the following assertion:
Let P be a point on the altitude AH of triangle ABC such that H is located between B and C. Extend BP and CP to their intersection with AC and AB in D and E, respectively. Then AH is the angle bisector of  DHE, or equivalently BHE = CHD. |
This is known as the Mirror Property of the altitudes and hence of the orthic triangle. (See also the mirror property in disguise proven using Pascal's theorem.) It was the main device in solving Fagnano's problem.
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Proof
The proof is by Professor Hiroshi Haruki.
Draw a line through A parallel to BC, and let F and G denote the intersections with the line of HE and HD extended. The assertion will follow from the fact that
By construction, triangles BEH and AEF are similar:
| AF = AE·BH / BE. |
Similarly,
| AG = AD·CH / CD. |
By Ceva's theorem we have
| AE/BE · BH/CH · CD/AD = 1, |
or,
| (BH·AE/BE)·(CD/(CH·AD)) = 1, |
which reduces to AF/AG = 1, yielding AF = AG.
References
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