Problem 4

Let

A,B

be distinct points on a given circle

O

, and let

P

be the midpoint of

AB

. Let

O_1

be the circle tangent to the line

AB

P

and tangent to

O

. Let

\ell

, different from

AB

, be the tangent to

O_1

through

A

, and let

C

be the intersection point other than

A

\ell

and

O

. Let

Q

be the midpoint of

BC

, and let

O_2

be the circle tangent to the line

BC

Q

and tangent to the line segment

AC

. Prove that

O_2

is tangent to

O

Step 1 of 5: Name the auxiliary points

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S=O\cap O_1,\quad T=SP\cap O,\quad X=\ell\cap O_1,\quad M=\operatorname{mid}(XP)

Detailed analysis

Let $S$ be the tangency point of $O$ and $O_1$ , let $T$ be the second intersection of $SP$ with $O$ , let $X$ be the tangency point of $\ell$ and $O_1$ , and let $M$ be the midpoint of $XP$ .