APMO

Question

In a circle C with centre O and radius r, let C_1, C_2 be two circles with centres O_1, O_2 and radii r_1, r_2 respectively, so that each circle C_i is internally tangent to C at A_i and so that C_1, C_2 are externally tangent to each other at A. Prove that the three lines O A, O_1 A_2, and O_2 A_1 are concurrent. FIGURE_0

Accepted Answer

Because of the tangencies, the following triples of points (two centers and a tangency point) are collinear:

O_{1}, O_{2}, A, O, O_{1}, A_{1}, O, O_{2}, A_{2} .

Because of that we can ignore the circles and only draw their centers and tangency points.

Now the problem is immediate from Ceva's theorem in triangle

O O_{1} O_{2}

, because

\frac{O A _{1}}{A _{1} O _{1}} \cdot \frac{O _{1} A}{A O _{2}} \cdot \frac{O _{2} A _{2}}{A _{2} O} = \frac{r}{r _{1}} \cdot \frac{r _{1}}{r _{2}} \cdot \frac{r _{2}}{r} = 1.