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 2 of 4: Use linear interpolation

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In plain words

A continuous linear quantity crosses zero between the endpoints precisely under the sign condition.

f(t)=(A-P)\cdot((1-t)(B-P)+t(C-P))=(1-t)f(0)+tf(1)

Detailed analysis

Define $f(t)$ to be the dot product. It is affine in $t$ , so some $t\in[0,1]$ satisfies $f(t)=0$ exactly when $f(0)$ and $f(1)$ have opposite signs or one is zero.