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Geometry - 2016 Taiwan TST Round 1 Quiz 1 Problem 2

Source: 2016 Taiwan TST Round 1 Quiz 1 Problem 2

March 29, 2016

geometry

Problem Statement

Circles

O_1

and

O_2

intersects at two points

B

and

C

, and

BC

is the diameter of circle

O_1

. Construct a tangent line of circle

O_1

at

C

and intersecting circle

O_2

at another point

A

. We join

AB

to intersect

O_1

at point

E

, then join

CE

and extend it to intersect circle

O_2

at point

F

. Assume that

H

is an arbitrary point on the line segment

D

. We join

HE

and extend it to intersect circle

O_1

at point

G

, and join

BG

and extend it to intersect the extended line of

AC

at point

D

. Prove that

\frac{AH}{HF}=\frac{AC}{CD}

.

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