HYACINTHOS 26862

[Antreas P. Hatzipolakis]:

 

 

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

Denote:

A"B"C" = the reflection of ABC in P

For P = O, N :
The circuncircles of HA'A", HB'B", HC'C" are coaxial.

2nd, other than H, intersections?

Locus of P?

 

 

[Angel Monesdeoca]:

 

 

**** The locus of P  such that the  circuncircles of HA'A", HB'B", HC'C" are coaxial is the cubic cgK376, complement of isogonal conjugate of K376 (Orthocubic's sister cubic  of catalogue CTC, Bernard Gibert).

Barycentric equation  of cgK376:

a^4 b^2 x^3-2 a^2 b^4 x^3+b^6 x^3-a^4 c^2 x^3+b^4 c^2 x^3+2 a^2 c^4 x^3-b^2 c^4 x^3-c^6 x^3+a^6 x^2 y-2 a^4 b^2 x^2 y+a^2 b^4 x^2 y+a^4 c^2 x^2 y-b^4 c^2 x^2 y-5 a^2 c^4 x^2 y-2 b^2 c^4 x^2 y+3 c^6 x^2 y-a^4 b^2 x y^2+2 a^2 b^4 x y^2-b^6 x y^2+a^4 c^2 x y^2-b^4 c^2 x y^2+2 a^2 c^4 x y^2+5 b^2 c^4 x y^2-3 c^6 x y^2-a^6 y^3+2 a^4 b^2 y^3-a^2 b^4 y^3-a^4 c^2 y^3+b^4 c^2 y^3+a^2 c^4 y^3-2 b^2 c^4 y^3+c^6 y^3-a^6 x^2 z-a^4 b^2 x^2 z+5 a^2 b^4 x^2 z-3 b^6 x^2 z+2 a^4 c^2 x^2 z+2 b^4 c^2 x^2 z-a^2 c^4 x^2 z+b^2 c^4 x^2 z+3 a^6 y^2 z-5 a^4 b^2 y^2 z+a^2 b^4 y^2 z+b^6 y^2 z-2 a^4 c^2 y^2 z-2 b^4 c^2 y^2 z-a^2 c^4 y^2 z+b^2 c^4 y^2 z-a^4 b^2 x z^2-2 a^2 b^4 x z^2+3 b^6 x z^2+a^4 c^2 x z^2-5 b^4 c^2 x z^2-2 a^2 c^4 x z^2+b^2 c^4 x z^2+c^6 x z^2-3 a^6 y z^2+2 a^4 b^2 y z^2+a^2 b^4 y z^2+5 a^4 c^2 y z^2-b^4 c^2 y z^2-a^2 c^4 y z^2+2 b^2 c^4 y z^2-c^6 y z^2+a^6 z^3+a^4 b^2 z^3-a^2 b^4 z^3-b^6 z^3-2 a^4 c^2 z^3+2 b^4 c^2 z^3+a^2 c^4 z^3-b^2 c^4 z^3=0.

Points on cgK376: A, B, C, midpoints of ABC,  circuncenter, orthocenter, center of the nine-point circle.

**** 2nd, other than H, intersections of circuncircles of HA'A", HB'B", HC'C":  

P=X(3)  -> W= X(4)X(64) /\ X(974)X(12004)

W = (a^2-b^2-c^2) (3 a^4-2 a^2 (b^2+c^2)-(b^2-c^2)^2)
     (2 a^16
      -a^14 (b^2+c^2)
     -13 a^12 (b^2-c^2)^2-(b^2-c^2)^8   
     +11 a^10 (b^2-c^2)^2 (b^2+c^2)
     +a^8 (b^2-c^2)^2 (35 b^4-54 b^2 c^2+35 c^4)
       -a^6 (b^2-c^2)^2 (67 b^6-51 b^4 c^2-51 b^2 c^4+67 c^6)
     +a^4 (b^2-c^2)^4 (41 b^4+94 b^2 c^2+41 c^4)
     -a^2 (b^2-c^2)^4 (7 b^6+41 b^4 c^2+41 b^2 c^4+7 c^6)    ) : .... : .....

On lines: {4,64}, {974,12004}

(6 - 9 - 13) - search numbers  of W: (7.05018113511390, 7.02917267360016, -4.47961558525291).


P=X(5) ->  X(2072) = inverse-in-nine-point-circle of X(3).


Angel Montesdeoca