HYACINTHOS 27498

[Antreas P. Hatzipolakis]:


Let ABC be a triangle P a point on the Euler line L and A'B'C' the pedal triangle of P.

Denote:

La, Lb, Lc = the reflections of L in PA', PB', PC', resp.

The parallels to La, Lb, Lc through A, B, C, resp. are concurrent at a
point U (independent from P)

The parallels to La, Lb, Lc though A', B', C', resp. are concurrent at a
point W.

Which is the locus of W as P moves on the Euler line ?

For which points P the W is lying on the NPC ?
(One is the H and the W is the antipode of X(125) in the NPC).


[Angel Montesdeoca]:


*** U = X(265)

*** W = midpoint of P and X(110)

*** Pairs {P,W}, whith P on the Euler line:

{2,5642}, {3,1511}, {4,113}, {5,10272}, {20,16163}, {21,16164},
{22,16165}, {23,1495}, {30,30}, {140,13392}, {382,1539}, {549,11694},
{858,11064}, {3146,13202}, {3153,1568}, {3524,11693}, {7464,10564},
{7471,3233}, {10296,1531}, {10989,13857}, {11050,15354}, {14807,14499},
{14808,14500}.

***
If P = Z, then W = X(3258) on the NPC.

Z =  a^12 -2 a^10 (b^2+c^2) +2 a^8 (b^4+b^2 c^2+c^4) -a^6 (3 b^6-b^4 c^2-b^2 c^4+3 c^6) +a^4 (2 b^8+b^6 c^2-5 b^4 c^4+b^2 c^6+2 c^8) +a^2 (b^2-c^2)^2 (b^6-3 b^4 c^2-3 b^2 c^4+c^6) -(b^2-c^2)^4 (b^4+b^2 c^2+c^4) : :

= lies on the lines: {2,3}, {110,3258}, {125,476}, {265,16168}, {477,15396}, {523,3448}, {542,14480}, {2777,14508}, {3265,13485},{6070,9140}, {9158,11645}, {12121,14851}, {12295,14989}, {12383,14934}, {14611,14683}, {14993,15027}.

= the midpoint of X(3448) and X(14731).

= the reflection of X(i) in X(j), for these {i, j}: {3,16340}, {110,3258}, {476,125}, {7471,3154}, {12383,14934}, {14683,14611}, {14989,12295}.

(6 - 9 - 13) - search numbers of Z: (7.00706237545655, 6.11905764508646, -3.82963498413235).

Angel Montesdeoca