HYACINTHOS 25302

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle and A'B'C' the pedal triangle of H.

Denote:

A1, B1, C1 = the reflections of O in BC, CA, AB, resp.

A2, B2, C2 = the reflections of A1, B1, C1 in B'C', C'A', A'B', resp.

1. The circumcircles of NA1A2, NB1B2, NC1C2 are coaxial.

 

2. The circumcircles of OA1A2, OB1B2, OC1C2 are coaxial.

Second points of intersections?

GENERALIZATION:

Let ABC be a triangle, A'B'C' the pedal triangle of H and P a point.

Denote:

A1, B1, C1 = the reflections of P in BC, CA, AB, resp.

A2, B2, C2 = the reflections of A1, B1, C1 in B'C', C'A', A'B', resp.

The circumcircles of PA1A2, PB1B2, PC1C2 are coaxial.

[Peter Moses]:

Hi Antreas,

 

1) X(23).

 

2) a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+6 a^6 b^2 c^2-2 a^4 b^4 c^2-8 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-2 a^4 b^2 c^4+12 a^2 b^4 c^4-2 b^6 c^4-8 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10::
on lines {{2,3},{11,4351},{12,4354},{ 94,10688},{113,511},{115,3003} ,{125,1533},{265,1177},{399, 3564},{495,9642},{524,5655},.. .}.

Midpoint of X(i) and X(j) for these {i,j}: {{4,23},{125,1533},{3581,7728} }.

Reflection of X(i) in X(j) for these {i,j}: {{3,468},{403,11563},{858,5},{ 2072,403},{7574,10297},{7728, 1514},{10295,7575},{10564, 5972}}.

 

>The circumcircles of P{p,q,r}A1A2, P{p,q,r}B1B2, P{p,q,r}C1C2 are coaxial.

(a^8 b^2-4 a^6 b^4+6 a^4 b^6-4 a^2 b^8+b^10+a^8 c^2-8 a^6 b^2 c^2+a^4 b^4 c^2+9 a^2 b^6 c^2-3 b^8 c^2-4 a^6 c^4+a^4 b^2 c^4-10 a^2 b^4 c^4+2 b^6 c^4+6 a^4 c^6+9 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10) p+(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-a^4 c^2-5 a^2 b^2 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6) q+(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2-5 a^2 b^2 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) r::

 

Best regards,

Peter Moses.