## Monday, November 23, 2009

### Problem 393: Triangle, Orthocenter, Circumcircles, Congruence, Collinear

Proposed Problem
Click the figure below to see the complete problem 393 about Triangle, Orthocenter, Circumcircles, Congruence, Collinear.

Complete Problem 393

Level: High School, SAT Prep, College geometry

#### 1 comment:

1. 1. Call O ,the center of circle 1. Call R, radius of circle 1
BD cut circle 1 at D’ .
AC is the perpendicular bisector of DD’ (property of orthocenter)
Triangle ADC is congruent to tri. AD’C (case SSS)
With the same logic, we will get radius of circles 1, 2,3,4,5 congruent.

2. Trapezoid ODD’G is isosceles so DG=OD’=R
Similarly DE=DF= R and D is the center of circle GEF

3. DE , DF and DG cut circles 2, 3 and 4 at H, M, N .
We have DH=DM=DN= 2R and circle 6 center D radius 2R will tangent to circles 2, 3, 4 at H,M,N

4. Angle (HBD)= angle(DBM)=90 so H,B,M is collinear .
similar logic will be applied for other points.

5. H is the intension of DE so D,E,H collinear.
similar logic will be applied for other points.

Peter Tran