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 1 of 6: Set the centers and the common chord

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In plain words

The radical axis of two circles is perpendicular to their centers line.

G_1G_2\perp AB

Detailed analysis

Let $O,O_1,O_2$ and $r,r_1,r_2$ be the centers and radii of $G,G_1,G_2$ . The common chord $AB$ is perpendicular to the line of centers $O_1O_2$ . Since $G_1$ passes through $O_2$ , we have $O_1O_2=r_1$ .