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 1 of 4: Parametrize the segment

In plain words

As $X$ slides along the segment, the perpendicularity expression changes linearly.

X=(1-t)B+tC,\quad 0\le t\le1,\quad (A-P)\cdot(X-P)=0

Detailed analysis

The right angle at $P$ is equivalent to perpendicular vectors $A-P$ and $X-P$ . Write every point of $BC$ as $X=(1-t)B+tC$ with $0\le t\le1$ .