# Geometric locus

1. Jun 16, 2017

### geoffrey159

1. The problem statement, all variables and given/known data

Given a general triangle ABC, find the geometric locus of points such that the three orthoprojection onto the sides of the triangle are aligned.

2. Relevant equations

Let's call A', B', and C' the orthoprojection of a given point M onto (AB) , (BC) , and (AC).
M satisfies the condition iff $(A'B',A'C') = 0\ (\mod \pi)$.

3. The attempt at a solution

It's easy to see that $MA'B'B$ and $MAC'A'$ are concyclic which translates into two equations mod $\pi$: $(A'B',A'M') = (BB',BM) ( = (BC,BM) )$ and $(A'M,A'C') = (AM,AC)$

Therefore, mod $\pi$, we have :
$(A'B',A'C') = 0 \iff (A'B',A'M) + (A'M,A'C) = 0 \iff (BC,BM) + (AM,AC) = 0 \iff (BC,BM) = (AC,AM)$

And we can conclude that A',B',C' are aligned iff M belongs to the circumscribed circle to ABC.

Is this correct ?

2. Jun 22, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Jun 23, 2017

### haruspex

Looks good, and rather neat.