Problem 2

Given a point

A

and a segment

BC

, determine the locus of all points

P

in space for which

\angle APX=90^\circ

for some point

X

on the segment

BC

Step 3 of 4: Identify the endpoint signs

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In plain words

The familiar right-angle sphere converts each endpoint dot product into a ball-membership test.

f(0)=(P-A)\cdot(P-B),\quad f(1)=(P-A)\cdot(P-C)

Detailed analysis

Changing both factors to $P-A$ gives the same dot products. By Thales' theorem, $(P-A)\cdot(P-B)\le0$ means that $P$ lies in the closed ball with diameter $AB$ ; similarly for $AC$ .