What is the geometric locus of triangle orthoprojections in alignment?

[geoffrey159]

Homework Statement

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

Homework 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 $(A'B',A'C') = 0\ (\mod \pi)$.

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 ?

 

[haruspex]

Homework Statement

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

Homework 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 $(A'B',A'C') = 0\ (\mod \pi)$.

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 ?

Looks good, and rather neat.