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 5 of 5: Identify the center and tangency point

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NY\perp AC,\quad NY=NQ=NA',\quad N\text{ is the center of }O_2

Detailed analysis

Let $Y$ be the tangency point of $O_2$ with $AC$ . Since $N,C,T$ are collinear, the angle relations in the construction give $\angle ACN=\angle ACT=\angle BCT=\angle QCN$ . Also $CY=CQ$ because the two segments are tangents from C to $O_2$ . Thus triangles $YCN$ and $QCN$ are congruent, so $NY\perp AC$ and $NY=NQ=NA'$ . Hence N is the center of $O_2$ . Finally, $O,A',Q$ lie on the perpendicular bisector of $BC$ , and $A'$ lies on both circles; the line of centers passes through $A'$ , proving that $O_2$ is tangent to $O$ at $A'$ .