Problem 4

Let

\mathcal C

be a circle tangent to a line

L

, and let

M\in L

. Find the locus of points

P

for which there are distinct points

Q,R\in L

equidistant from

M

such that

\mathcal C

is the incircle of triangle

PQR

Step 3 of 4: Identify the auxiliary touchpoint

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In plain words

Identify the auxiliary touchpoint

QY'\,XQ=OX\,O'Y'=RX\,Y'R\quad\Longrightarrow\quad QX=RY'

Detailed analysis

Multiplying the two ratios gives $QY'\cdot XQ=RX\cdot Y'R$ . Since the two decompositions of QR agree, this yields $QX=RY'$ . Together with $QX=RY$ , we get $Y=Y'$ . Hence P lies on line YZ, and the incircle condition selects the open ray from Z away from Y.