HYACINTHOS 28997

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, P. Q two isogonal conjugate points and A'B'C', A"B"C" the pedal triangles of P,Q, resp.

Denote:

A* = B'C" /\ B"C'
B* = C'A" /\ C"A'
C* = A'B" /\ A"B'

 

Which is the locus of P such that A*, B*, C* are collinear?
The entire plane?
O, H lie on the locus 
(Stathis Kutras, Romantics of Geometry 2980)

Which is the centroid of the (degenerated) triangle A*B*C*?

Special cases:
1. P = O or H
2. P = G or K

 

[Peter Moses]:

Hi Antreas,

 

> Which is the centroid of the (degenerated) triangle A*B*C*? 
Hyacinthos 26084 

>1. P = O or H
X(13448).

>2. P = G or K

4 a^12-21 a^10 b^2-87 a^8 b^4-94 a^6 b^6-30 a^4 b^8+3 a^2 b^10+b^12-21 a^10 c^2+324 a^8 b^2 c^2-9 a^6 b^4 c^2+546 a^4 b^6 c^2-72 a^2 b^8 c^2-87 a^8 c^4-9 a^6 b^2 c^4-954 a^4 b^4 c^4+93 a^2 b^6 c^4-9 b^8 c^4-94 a^6 c^6+546 a^4 b^2 c^6+93 a^2 b^4 c^6-16 b^6 c^6-30 a^4 c^8-72 a^2 b^2 c^8-9 b^4 c^8+3 a^2 c^10+c^12 : : 

 

= lies on the line: {2,6}.

Best regards,
Peter Moses.