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 2 of 4: Compute the three directed side ratios

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\frac{OA_1}{A_1O_1}=\frac{r}{r_1},\qquad\frac{O_1A}{AO_2}=\frac{r_1}{r_2},\qquad\frac{O_2A_2}{A_2O}=\frac{r_2}{r}

Detailed analysis

The internal tangencies give $OA_1=r$ and $A_1O_1=r_1$ , and similarly $O_2A_2=r_2$ and $A_2O=r$ . The external tangency gives $O_1A=r_1$ and $AO_2=r_2$ , so the displayed ratios follow (with the usual directed-segment interpretation for the points on extensions).