HYACINTHOS 29543

 

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.

N'a, N'b, N'c = the reflections of Na, Nb, Nc in BC, CA, AB, resp.

Ga, Gb, Gc = the centroids of NaN'bN'c, NbN'cN'a, NcN'aN'b, resp.
G'a, G'b, G'c = the centroids of N'aNbNc,N'bNcNa, N'cNaNb, resp.

1. The NPC centers of GaGbGc and G'aG'bG'c lie on the Euler line of ABC.
They are symmetric in the centroid G of ABC.
Points ?

 

2. (Ga, G'a), (Gb, G'b), (Gc, G'c) are symmetric pairs in G
(ie G is the midpoint of GaG'a, GbG'b, GcG'c)
Therefore  Ga, Gb, Gc, G'a, G'b, G'c lie on a conic centered at G.
Perspector of the conic?


[Peter Moses]:


Hi Antreas,

1).
-----------------------------------------------------

2*a^10 - 6*a^8*b^2 + 4*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^8 + 2*b^10 - 6*a^8*c^2 + 6*a^6*b^2*c^2 - 3*a^4*b^4*c^2 + 9*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 - 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 4*b^6*c^4 + 4*a^4*c^6 + 9*a^2*b^2*c^6 + 4*b^4*c^6 - 6*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :

= 3 X[2] + X[10201], X[5] + 2 X[10125], 2 X[5] + X[15331], X[26] + 11 X[5070], 4 X[140] - X[10226], 2 X[140] + X[13406], 2 X[546] + X[15332], 2 X[547] + X[15330], 5 X[632] - 2 X[5498], 5 X[632] + 4 X[12010], 5 X[1656] + X[1658], 7 X[3526] - X[11250], 2 X[3628] + X[10020], 4 X[3628] - X[10224], 8 X[3628] + X[12107], X[5498] + 2 X[12010], 2 X[10020] + X[10224], 4 X[10020] - X[12107], 4 X[10125] - X[15331], 4 X[10212] - 7 X[14869], 2 X[10224] + X[12107], X[10226] + 2 X[13406], X[14070] + 7 X[15703]

= lies on these lines: {2, 3}, {141, 34155}, {5891, 16223}, {10182, 30522}, {18475, 20304}

= midpoint of X(10154) and X(13371)
-----------------------------------------------------

2*a^10 - 6*a^8*b^2 + 4*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^8 + 2*b^10 - 6*a^8*c^2 + 10*a^6*b^2*c^2 - 5*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 - 5*a^4*b^2*c^4 - 2*a^2*b^4*c^4 + 4*b^6*c^4 + 4*a^4*c^6 + 7*a^2*b^2*c^6 + 4*b^4*c^6 - 6*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :

= 5 X[2] - X[10201], X[5] + 2 X[5498], 2 X[5] + X[10226], 2 X[140] + X[10224], 4 X[140] - X[15331], X[550] - 4 X[10212], 5 X[632] - 2 X[10125], 5 X[632] - 8 X[12043], 10 X[632] - X[12107], 5 X[632] + X[13371], 5 X[1656] + X[11250], X[1658] - 7 X[3526], 4 X[3530] - X[15332], 4 X[3628] - X[13406], 11 X[5070] + X[12084], 4 X[5498] - X[10226], 4 X[10124] - X[15330], X[10125] - 4 X[12043], 4 X[10125] - X[12107], 2 X[10125] + X[13371], 2 X[10224] + X[15331], 16 X[12043] - X[12107], 8 X[12043] + X[13371], X[12107] + 2 X[13371], X[32171] + 2 X[32767]

= lies on these lines: {2, 3}, {49, 9140}, {541, 25563}, {542, 13561}, {567, 15059}, {575, 6698}, {6101, 13857}, {9730, 34128}, {10263, 32225}, {11430, 20304}, {11564, 15027}, {32171, 32767}

-----------------------------------------------------

2).
(4*a^20 - 24*a^18*b^2 + 53*a^16*b^4 - 38*a^14*b^6 - 41*a^12*b^8 + 92*a^10*b^10 - 41*a^8*b^12 - 38*a^6*b^14 + 53*a^4*b^16 - 24*a^2*b^18 + 4*b^20 - 24*a^18*c^2 + 105*a^16*b^2*c^2 - 164*a^14*b^4*c^2 + 102*a^12*b^6*c^2 - 19*a^10*b^8*c^2 - 19*a^8*b^10*c^2 + 102*a^6*b^12*c^2 - 164*a^4*b^14*c^2 + 105*a^2*b^16*c^2 - 24*b^18*c^2 + 52*a^16*c^4 - 162*a^14*b^2*c^4 + 178*a^12*b^4*c^4 - 94*a^10*b^6*c^4 + 52*a^8*b^8*c^4 - 94*a^6*b^10*c^4 + 178*a^4*b^12*c^4 - 162*a^2*b^14*c^4 + 52*b^16*c^4 - 32*a^14*c^6 + 91*a^12*b^2*c^6 - 95*a^10*b^4*c^6 + 36*a^8*b^6*c^6 + 36*a^6*b^8*c^6 - 95*a^4*b^10*c^6 + 91*a^2*b^12*c^6 - 32*b^14*c^6 - 56*a^12*c^8 + 4*a^10*b^2*c^8 + 69*a^8*b^4*c^8 + 38*a^6*b^6*c^8 + 69*a^4*b^8*c^8 + 4*a^2*b^10*c^8 - 56*b^12*c^8 + 112*a^10*c^10 - 41*a^8*b^2*c^10 - 122*a^6*b^4*c^10 - 122*a^4*b^6*c^10 - 41*a^2*b^8*c^10 + 112*b^10*c^10 - 56*a^8*c^12 + 110*a^6*b^2*c^12 + 192*a^4*b^4*c^12 + 110*a^2*b^6*c^12 - 56*b^8*c^12 - 32*a^6*c^14 - 163*a^4*b^2*c^14 - 163*a^2*b^4*c^14 - 32*b^6*c^14 + 52*a^4*c^16 + 104*a^2*b^2*c^16 + 52*b^4*c^16 - 24*a^2*c^18 - 24*b^2*c^18 + 4*c^20)*(4*a^20 - 24*a^18*b^2 + 52*a^16*b^4 - 32*a^14*b^6 - 56*a^12*b^8 + 112*a^10*b^10 - 56*a^8*b^12 - 32*a^6*b^14 + 52*a^4*b^16 - 24*a^2*b^18 + 4*b^20 - 24*a^18*c^2 + 105*a^16*b^2*c^2 - 162*a^14*b^4*c^2 + 91*a^12*b^6*c^2 + 4*a^10*b^8*c^2 - 41*a^8*b^10*c^2 + 110*a^6*b^12*c^2 - 163*a^4*b^14*c^2 + 104*a^2*b^16*c^2 - 24*b^18*c^2 + 53*a^16*c^4 - 164*a^14*b^2*c^4 + 178*a^12*b^4*c^4 - 95*a^10*b^6*c^4 + 69*a^8*b^8*c^4 - 122*a^6*b^10*c^4 + 192*a^4*b^12*c^4 - 163*a^2*b^14*c^4 + 52*b^16*c^4 - 38*a^14*c^6 + 102*a^12*b^2*c^6 - 94*a^10*b^4*c^6 + 36*a^8*b^6*c^6 + 38*a^6*b^8*c^6 - 122*a^4*b^10*c^6 + 110*a^2*b^12*c^6 - 32*b^14*c^6 - 41*a^12*c^8 - 19*a^10*b^2*c^8 + 52*a^8*b^4*c^8 + 36*a^6*b^6*c^8 + 69*a^4*b^8*c^8 - 41*a^2*b^10*c^8 - 56*b^12*c^8 + 92*a^10*c^10 - 19*a^8*b^2*c^10 - 94*a^6*b^4*c^10 - 95*a^4*b^6*c^10 + 4*a^2*b^8*c^10 + 112*b^10*c^10 - 41*a^8*c^12 + 102*a^6*b^2*c^12 + 178*a^4*b^4*c^12 + 91*a^2*b^6*c^12 - 56*b^8*c^12 - 38*a^6*c^14 - 164*a^4*b^2*c^14 - 162*a^2*b^4*c^14 - 32*b^6*c^14 + 53*a^4*c^16 + 105*a^2*b^2*c^16 + 52*b^4*c^16 - 24*a^2*c^18 - 24*b^2*c^18 + 4*c^20) : :

Best regards,
Peter Moses.