## Thursday, September 26, 2013

### Geometry Problem 929: Triangle, Orthocenter, Circumcircle, Reflection, Concurrency

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 929.

1. http://img43.imageshack.us/img43/9618/h2bb.png
Let BH,AH and CH cut the circle at B’, A’ and C’
Note that A’, B’ and C’ is the reflection of H over BC, AC and AB ..( property of orthocenter)
So F, B’, B1 are collinear , similarly for E, A’, A1 and D, C’, C1
This problem become problem 733 . see below for solution
http://gogeometry.blogspot.com/2012/02/problem-733-triangle-orthocenter.html

2. Let A', B', and C' be reflections of H over BC, AC, and AB respectively. By property of orthocenter, these points are on circumcircle.
Let FB' and DC' meet at P.
<FPD=180-2(<DFA+<FDA)=180-2<BAC=arcB'C'/2, making P on circumcircle.
Let DC' and EA' meet at P'.
<DP'E=180-(180-2<BDE)-(180-2<BED)=2(<BDE+<BED)-180=2(180-<ABC)-180=180-2<ABC
=arcA'C'/2, making P' on circumcircle.
P and P' are both on DC' and circumcircle, so they must coincide.