WOOT 2014-15: Geometry of the Circle A

1. Two circles have common external tangents AB and CD. A common internal tangent intersects AB at P andCD at Q. Prove that PQ = AB.

2. Let ABCD be a cyclic quadrilateral. Perpendicular diagonals AC and BD intersect at X . Prove that the linethrough X perpendicular to AD passes through the midpoint of BC.

3. A convex pentagon inscribed in a circle has side lengths 16, 16, 16, 16, and 6. What is its area?

4. Points A, B, C, and D are consecutive vertices of a regular n-gon and 1

AB= 1

AC+ 1

AD. Determine n.

5. Let ABC be an acute triangle, and let D, E, and F be points on sides BC, AC, and AB, respectively, sothat CD/CE = CA/CB, AE/AF = AB/AC, and BF/BD = BC/BA. Prove that AD, BE, and CF are thealtitudes of triangle ABC.

6. ABCD is a convex quadrilateral with DA = DB = DC and AD ‖ BC. The perpendicular bisector of CDmeets AB at E such that B is between A and E. If ∠BCE = 2∠CED, compute ∠BCE.

7. Circles Γ1 and Γ2, with centers O1 and O2, respectively, are externally tangent. Let PQ be a commonexternal tangent of Γ1 and Γ2, with P on Γ1 and Q on Γ2. Let Γ3 be the circle with diameter O1O2. Provethat PQ is also tangent to Γ3.

8. Let ABC be a triangle, and I its incenter. Let the incircle of triangle ABC touch side BC at D, and let linesBI and CI meet the circle with diameter AI at points P and Q, respectively. Given BI = 6, CI = 5,DI = 3, determine the value of (DP/DQ)2.

9. Let AB be a chord of a circle centered at O. Let ON be the radius perpendicular to AB, meeting AB at M .Let P be any point on major arc AB, not diametrically opposite N . Let PM intersect the circle again at Q,and let PN intersect AB at R. Prove that RN > MQ.

A BM

N

O

P

Q

R

10. Let r be the inradius of triangle ABC. Let r1 be the radius of the circle that is tangent to AB, AC, and theincircle, as shown. Define r2 and r3 similarly. Show that r =

√r1r2 +

√r1r3 +

√r2r3.

A

B C