HYACINTHOS 25540

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

La = the perpendicular to AI at I.

The perpendicular to AB at B intersects La at A2

The perpendicular to AC at C intersects La at A3

The perpendiculars to La at A2, A3 intersect BC at Ab, Ac respectively.

Smilarly Bc, Ba and Ca, Cb.

The six points Ab, Ac, Bc, Ba, Ca, Cb lie on a conic (*)

Complicated construction! :)

Equivalently:

Let ABC be a triangle and A'B'C' the antipedal triangle of I.

The parallel through C' to AI intersects BC at Ab

The parallel through B' to AI intersects BC at Ac.

Smilarly Bc, Ba and Ca, Cb.

The six points lie on a conic.

 

Question:
For which other than I points P (A'B'C' = antipedal triangle of a point P) the six points lie on a conic?

(*)

Angel Montesdeoca, El punto X(1630) como centro radical

[Angel Montesdeoca]:

> For which other than I points P (A'B'C' = antipedal

triangle of a point P) the six points lie on a conic?

*** If P lies on a nonic (complicated equation) that passes through the vertices of the triangles ABC and excentral, by the incenter and the point of De Longchamps.

 

[APH]

 

Is the center of the conic for P = de Longchamps point as simple as it is for P = I ?

(W = ( a(a+b-c)(a-b+c)(b+c) .... etc )

[Angel Montesdeoca]:


Dear Antreas


The coordinates of the center of the conic for P = de Longchamps point  are complicated. The first barycentric coordinate is:

3 (b^2-c^2)^8 (25 b^8+188 b^6 c^2+342 b^4 c^4+188 b^2 c^6+25 c^8)
-2 (b^2-c^2)^6 (167 b^10+123 b^8 c^2-1826 b^6 c^4-1826 b^4 c^6+123 b^2 c^8+167 c^10) a^2
+4 (b^2-c^2)^4 (69 b^12-1094 b^10 c^2-277 b^8 c^4+3628 b^6 c^6-277 b^4 c^8-1094 b^2 c^10+69 c^12) a^4
+2 (b^2-c^2)^4 (467 b^10+2599 b^8 c^2-7930 b^6 c^4-7930 b^4 c^6+2599 b^2 c^8+467 c^10) a^6
-(b^2-c^2)^2 (2073 b^12-9110 b^10 c^2-12937 b^8 c^4+44044 b^6 c^6-12937 b^4 c^8-9110 b^2 c^10+2073 c^12) a^8
+4 (b^2-c^2)^2 (165 b^10-3599 b^8 c^2+5098 b^6 c^4+5098 b^4 c^6-3599 b^2 c^8+165 c^10) a^10
+32 (b^2-c^2)^2 (63 b^8+68 b^6 c^2-782 b^4 c^4+68 b^2 c^6+63 c^8) a^12
-4 (b^2-c^2)^2 (549 b^6-1877 b^4 c^2-1877 b^2 c^4+549 c^6) a^14
+(201 b^8-5252 b^6 c^2+9846 b^4 c^4-5252 b^2 c^6+201 c^8) a^16
+22 (39 b^6-23 b^4 c^2-23 b^2 c^4+39 c^6) a^18
-4 (125 b^4-98 b^2 c^2+125 c^4) a^20
+78 (b^2+c^2) a^22
+5 a^24

with (6-9-13)-search numbers (2.00137610678924, 1.52349876340136, 1.66222251949610).

Angel Montesdeoca