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 2 of 4: Compare the two tangent circles

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In plain words

Compare the two tangent circles

\dfrac{QY'}{Y'O'}=\dfrac{OX}{XQ},\qquad \dfrac{RX}{XO}=\dfrac{O'Y'}{Y'R}

Detailed analysis

Let C-prime be the circle on the opposite side of QR, tangent to PQ, PR, and QR, with touchpoint Y-prime and center O-prime. The right-triangle similarities at X and Y-prime give the two ratios displayed above.