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 3 of 5: Introduce the perpendicular-bisector point

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A'\in O,\quad A'Q\perp BC,\quad N=A'Q\cap CT

Detailed analysis

Let $A'$ be the point of $O$ on the perpendicular bisector of $BC$ lying on the same side of $BC$ as $A$ , and put $N=A'Q\cap CT$ . The angle equalities in the official construction give $\triangle NCQ\sim\triangle TBP$ and $\triangle CA'Q\sim\triangle SPX$ .