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 4 of 4: Prove the converse and state the locus

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In plain words

Prove the converse and state the locus

P\in\{Z+t(Y-Z):t<0\}

Detailed analysis

Conversely, take P on the open ray from Z away from Y. Reversing the argument gives $QX=RY$ . Since M is the midpoint of XY, it is also the midpoint of QR; suitable Q,R therefore exist and C is the incircle.