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 6 of 6: Conclude tangency

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In plain words

A line is tangent exactly when its center distance equals the radius.

x=r_2

Detailed analysis

Substitute the expression for $a$ into the distance relation. It simplifies to $x=r_2$ . Thus the distance from the center $O_2$ of $G_2$ to line $CD$ equals the radius $r_2$ , so $CD$ is tangent to $G_2$ .