Problem 2

In a circle

C

with centre

O

and radius

r

, let

C_1,C_2

have centres

O_1,O_2

and radii

r_1,r_2

. Each

C_i

is internally tangent to

C

A_i

, and

C_1,C_2

are externally tangent at

A

. Prove that the lines

OA

O_1A_2

, and

O_2A_1

are concurrent.

Step 1 of 4: Record the three collinearities forced by tangency

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O,O_1,A_1\text{ are collinear},\quad O,O_2,A_2\text{ are collinear},\quad O_1,O_2,A\text{ are collinear}

Detailed analysis

At an internal tangency, the two centres and the tangency point are collinear; at an external tangency, the two centres and the tangency point are collinear as well. Thus $A_1$ lies on $OO_1$ , $A_2$ lies on $OO_2$ , and $A$ lies on $O_1O_2$ .