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 2 of 6: Use the tangency homothety at

M

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In plain words

Tangency creates a dilation from the inner circle to the outer one.

CD\parallel AB

Detailed analysis

The homothety centered at $M$ taking $G_1$ to $G$ sends $C,D$ to $A,B$ , because $M,C,A$ and $M,D,B$ are collinear. Therefore $CD\parallel AB$ , so $CD\perp O_1O_2$ .