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 4 of 5: Show that N is a midpoint

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\frac{QN}{QC}=\frac{PT}{PB},\quad\frac{QC}{QA'}=\frac{XS}{XP}\Longrightarrow QA'=2QN

Detailed analysis

The two similarities give $\frac{QN}{QC}=\frac{PT}{PB}$ and $\frac{QC}{QA'}=\frac{XS}{XP}$ . Their product, together with $\frac{XS}{XP}\cdot\frac{PT}{PB}=\frac12$ , gives $\frac{QN}{QA'}=\frac12$ , so $QA'=2QN$ and $N$ is the midpoint of $QA'$ .