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 3 of 4: Apply Ceva's theorem in the triangle of centres

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\frac{OA_1}{A_1O_1}\cdot\frac{O_1A}{AO_2}\cdot\frac{O_2A_2}{A_2O}=\frac{r}{r_1}\cdot\frac{r_1}{r_2}\cdot\frac{r_2}{r}=1

Detailed analysis

In triangle $OO_1O_2$ , the cevians are $O_2A_1$ , $OA$ , and $O_1A_2$ . The product of their three directed side ratios is $1$ , exactly the condition in Ceva's theorem; therefore these three lines are concurrent.