Problem 5

Two circles

G_1

and

G_2

are contained inside the circle

G

, and are tangent to

G

at the distinct points

M

and

N

, respectively.

G_1

passes through the center of

G_2

. The line passing through the two points of intersection of

G_1

and

G_2

meets

G

A

and

B

. The lines

MA

and

MB

meet

G_1

C

and

D

, respectively. Prove that

CD

is tangent to

G_2

Step 4 of 6: Compute the distance to

CD

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In plain words

Translate the parallel chords through the tangency homothety.

(r_1-x)=\frac{r_1}{r}\left(a-\frac{r_2^2}{2r_1}\right)

Detailed analysis

Use coordinates with origin $O_2$ , $x$ -axis $O_2O_1$ , and $O=(a,b)$ . Let $W=CD\cap O_1O_2$ and $O_2W=x$ . The homothety at $M$ has ratio $r/r_1$ , so the distance from $O_1$ to $CD$ is related to the distance from $O$ to $AB$ by the displayed equation.