MathLabs

Problem 4

Given a plane π\pi, a point PP in π\pi and a point QQ not in π\pi, find all points RR in π\pi such that the ratio QP+PRQR\dfrac{QP+PR}{QR} is a maximum.
Step 3 of 5: Convert the numerator into a distance
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S∈π,PS=PQ,S opposite X across PS\in\pi,\quad PS=PQ,\quad S\text{ opposite }X\text{ across }P
Detailed analysis

When P≠X, choose S on the line PX on the side opposite X with PS=PQ. For R on the ray PX, the triangle inequality is an equality in SR=SP+PR=QP+PR.