HYACINTHOS 29631

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the medial triangle.

Denote:

A", B", C" = the midpoints of AO, BO, CO, resp.

Oa, Ob, Oc = the circumcenters of OBC, OCA, OAB, resp.
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.

Goa, Gob, Goc = the centroids of A'A"Oa, B'B"Ob, C'C"Oc, resp.  
Gna, Gnb, Gnc = the centroids of A'A"Na, B'B"Nb, C'C"Nc, resp.

1. The circumcenter of GoaGobGoc lies on the Euler line
2. The NPC center of GnaGnbGnc lies on the Euler line.

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Let ABC be a triangle and A'B'C' the medial triangle.

Denote:

A", B", C"  the midpoints of AN, BN, CN, resp.

Oa, Ob, Oc = the circumcenters of NBC, NCA, NAB, resp.

Ga, Gb, Gc = the centroids of OaA'A", ObB'B", OcC'C", resp.

3. The NPC center of GaGbGc lies on the Euler line


[Peter Moses]:


Hi Antreas,

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1).
= 4*a^10 - 9*a^8*b^2 + 2*a^6*b^4 + 8*a^4*b^6 - 6*a^2*b^8 + b^10 - 9*a^8*c^2 + 12*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 6*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 6*a^4*b^2*c^4 + 2*b^6*c^4 + 8*a^4*c^6 + 6*a^2*b^2*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 : :

= lies on these lines: {2, 3}, {343, 1511}, {2883, 32210}, {3917, 16223}, {5642, 11562}, {5663, 10192}, {6697, 11645}, {6699, 20773}, {10182, 13754}, {10193, 14915}, {10282, 20191}, {12359, 32171}, {16226, 32352}, {16252, 32138}, {17821, 32140}

= midpoint of X(i) and X(j) for these {i,j}: {2,18324}, {3,10201}, {549,34351}, {14070,18281}, {15331,34330}
= reflection of X(i) in X(j) for these {i,j}: {5,34330}, {10201,10020}, {15761,10201}, {18566,5066}, {34330,10125}, {34351,15330}

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2).
= 6*a^10 - 15*a^8*b^2 + 6*a^6*b^4 + 12*a^4*b^6 - 12*a^2*b^8 + 3*b^10 - 15*a^8*c^2 + 26*a^6*b^2*c^2 - 13*a^4*b^4*c^2 + 11*a^2*b^6*c^2 - 9*b^8*c^2 + 6*a^6*c^4 - 13*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 6*b^6*c^4 + 12*a^4*c^6 + 11*a^2*b^2*c^6 + 6*b^4*c^6 - 12*a^2*c^8 - 9*b^2*c^8 + 3*c^10 : :

= lies on these lines: {2, 3}, {3410, 22251}

= midpoint of X(i) and X(j) for these {i,j}: {549,34331}, {15330,18281}, {15332,18568}
= reflection of X(25401) in X(34331) 

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3).
= 2*a^16 - 17*a^14*b^2 + 65*a^12*b^4 - 139*a^10*b^6 + 175*a^8*b^8 - 127*a^6*b^10 + 47*a^4*b^12 - 5*a^2*b^14 - b^16 - 17*a^14*c^2 + 90*a^12*b^2*c^2 - 171*a^10*b^4*c^2 + 98*a^8*b^6*c^2 + 89*a^6*b^8*c^2 - 138*a^4*b^10*c^2 + 51*a^2*b^12*c^2 - 2*b^14*c^2 + 65*a^12*c^4 - 171*a^10*b^2*c^4 + 96*a^8*b^4*c^4 + 29*a^6*b^6*c^4 + 72*a^4*b^8*c^4 - 123*a^2*b^10*c^4 + 32*b^12*c^4 - 139*a^10*c^6 + 98*a^8*b^2*c^6 + 29*a^6*b^4*c^6 + 38*a^4*b^6*c^6 + 77*a^2*b^8*c^6 - 94*b^10*c^6 + 175*a^8*c^8 + 89*a^6*b^2*c^8 + 72*a^4*b^4*c^8 + 77*a^2*b^6*c^8 + 130*b^8*c^8 - 127*a^6*c^10 - 138*a^4*b^2*c^10 - 123*a^2*b^4*c^10 - 94*b^6*c^10 + 47*a^4*c^12 + 51*a^2*b^2*c^12 + 32*b^4*c^12 - 5*a^2*c^14 - 2*b^2*c^14 - c^16 : :

= lies on this line: {2, 3}.

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Best regards,
Peter Moses.