## Wednesday, October 21, 2009

### Problem 370. Triangle with squares, Circumcircles, Tangent circles

Proposed Problem
Click the figure below to see the complete problem 370 about Triangle with squares, Circumcircles, Tangent circles. See more:
Complete Geometry Problem 370
Level: High School, SAT Prep, College geometry

1. http://img69.imageshack.us/img69/1681/problem3701.png
O1, O2, O3 and O4 are the centers of circles 1,2,3,4
R1, R2, R3, R4 are radius of circles 1,2,3,4
Triangle (ABC) is similar to triangle (A4BC4) because (AC//A4C4)
Ratio of similarity = BA/BA4= R1/R4 =AO1/A4O4= BO1/BO4
Consider 2 triangles AO1B and A4O4B,
We have BA/BA4=AO1/A4O4=BO1/BO4= R1/R4 so Tri(AO1B) is similar to tri (A4O4B)
So AO1//A4O4 and angle (ABO1)=angle(A4BO4) and B, O1, O4 is collinear.

Since B, O1,O4 are collinear,
Circle 1, center O1, Radius O1B will tangent to circle 4 , center O4, and radius O4B at B.

With similar reasons, Circle1 will be tangent to circles 2 and 3 at C and A.

Peter Tran

2. Let l1 be the tangent to circle 4 at B.
∠CAJ + ∠AJH = 90° + 90° = 180°
∴ CA // HJ
∠BAC = ∠BA4C4
∠ABC = ∠A4BC4
∴ ΔABC ~ ΔA4BC4
∴ ∠ACB = ∠A4C4B
∵ l1 be the tangent to circle 4 at B.
∴ ∠A4C4B = angle between chord C4B and l1.
∴ ∠ACB = angle between chord CB and l1.
∵ O1B⟂l1 and O4B⟂l1
∴ O1B // O4B
∴ O1O4B collinear
∴ O1O4 + O1B = O4B
O1O4 = O4B - O1B
∴O1 tangent to O4.
Similarly, O1 will be tangent to O2 and O3