stan@hare.DEC (Stanley Rabinowitz) (08/07/84)
I too, am an avid collector of geometry problems. I have hundreds of
problems in my collection. (I guess I've just been collecting longer
than ecsvax!pizer.)
My idea of an elegant geometry problem is one that is easy to state,
but relatively hard to prove. Here now are 5 "hard" problems from
my collection for your enjoyment:
1. In triangle ABC, AB=AC. M is the midpoint of BC. Point D is on AC
such that MD is perpendicular to AC. E is the midpoint of MD.
Prove that AE is perpendicular to BD.
2. In triangle ABC, P is any point on altitude AH. BP meets AC at D.
CP meets AB at E. Prove that angle EHA = angle DHA.
3. The circle inscribed in triangle ABC touches side BC at point D.
DE is a diameter of this circle. AE meets BC at F. Prove BD=FC.
4. ABCD is a parallelogram. E is any point on side AD. F is any point on
side BC. BE meets AF at P. CE meets DF at Q. PQ (extended) meets AB at R
and meets CD at S. Prove AR=CS.
5. AB and BC are two chords of a circle (with B lying on minor arc AC) with
BC > AB. Let M be the midpoint of arc ABC. Let D be the foot of the
perpendicular dropped from M to chord BC. Prove AB+BD=CD.
[This is known as the Archimedes Broken Chord Theorem.]
enjoy, Stanley Rabinowitz
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