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 5 of 6: Use the two tangencies to

G

In plain words

The two distance equations determine the needed coordinate.

a=\frac{r_2^2}{2r_1}+r-\frac{rr_2}{r_1}

Detailed analysis

Internal tangency gives $(r-r_1)^2=(r_1-a)^2+b^2$ and $(r-r_2)^2=a^2+b^2$ . Subtracting eliminates $b$ and yields the displayed expression for $a$ .