## 22495Re: Reflections in altitudes - Locus

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• Jul 3, 2014

[APH]

Let ABC be a triangle and P a point.

Denote:

P1 = the reflection of P in AH
Pa = the reflection of P1 in BC
Ppa = the reflection of Pa in PA

Ma = the midpoint of PPpa

Similarly Mb, Mc

Which is the locus of P such that ABC, MaMbMc are perspective?

If P = I, then they are, with perspector the circumcenter O.
Also for P = N the triangles are perspective.
[the congruent circles (Nna) and (N) are tangent,
so Ma is the tangency point. Where:
(N1) = the reflection of the NPC (N) in AH
(Na) = the reflection of (N1) in BC
(Nna) = the reflection of (Na) in NA)]

In fact Ppa, Ppb, Ppc  are the reflections of the feet of
altitudes in PA, PB, PC, resp.

So the proplem is:

Which is the locus of P such that the reflections of
the altitudes AA',BB',CC' in AP, BP, CP, resp. are concurrent?
(I think it was discussed in hyacinthos)

APH

For P = N:

The reflections of AH,BH,CH in AN,BN,CN are concurrent
APH
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