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 1 of 4: Encode the midpoint condition

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In plain words

Encode the midpoint condition

QX=RY

Detailed analysis

Because M is the midpoint of QR and XY, label Q,R so Q and X are on the same side of M. Then $QX=RY$ . We compare this with the second touchpoint of an auxiliary circle.