Problem 4

A semicircular arc

\gamma

has diameter

AB

. Let

C

be a point of the arc other than

A,B

, and let

D

be the foot of the perpendicular from

C

AB

. Three circles

\gamma_1,\gamma_2,\gamma_3

are tangent to line

AB

;

\gamma_1

is inscribed in triangle

ABC

, while

\gamma_2

and

\gamma_3

are tangent to

CD

and to

\gamma

, on opposite sides of

CD

. Prove that the three circles have a second common tangent.

Step 3 of 4: Locate the midpoint on the base

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In plain words

The base tangency points are arranged symmetrically around the incircle's tangency point.

XP=PY=r_1,\qquad XY=r_2+r_3=2r_1

Detailed analysis

Using $AD=b^2/c$ and the tangent-length formula, one obtains $XP=r_1$ ; symmetrically $PY=r_1$ . The previous step gives $XY=2r_1$ , so $P$ is the midpoint of $XY$ .