HYACINTHOS 26685

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of O.

Denote:

Ab, Ac = the midpoints of AB', AC', resp.
Bc, Ba = the midpoints of BC', BA', resp.
Ca, Cb = the midpoints of CA', CB', resp.

O1, O2 = the circumcenters of AbBcCa, AcCbBa, resp. (bicentric points)

The midpoint of O1O2 lies on the Euler line of ABC.

 

 

[Peter Moses]:

Hi Antreas,

8 a^4-11 a^2 b^2+3 b^4-11 a^2 c^2-6 b^2 c^2+3 c^4:: 

on lines {{2,3},{182,3631},{499,10386}, {575,3629},{576,6329},{590,645 4},{615,6453},{1385,3626},{148 3,3632},{1484,6154},{1587,6522 },{1588,6519},{2883,10193},{ 3055,5206},{3244,5690},{3564, 10541},{3592,5420},{3594,5418} ,{3636,6684},{3746,5433},{ 3819,6102},{3917,12006},{4031, 6147},{5010,7294},{5050,11008} ,{5092,5944},{5326,7280},{ 5432,5563},{5447,5946},{5609, 6699},{5650,11591},{5892,6101} ,{5901,7991},{6221,13993},{ 6247,10182},{6398,13925},{6407 ,13939},{6408,13886},{6419, 13966},{6420,8981},{6425,7584} ,{6426,7583},{6427,9540},{ 6428,13935},{7843,9771},{7863, 11168},{7982,10283},{9143, 13393},{9680,13847},{10170, 13491},{10263,11695},{10519, 11482},{10625,11592},{13372, 14073}}.

complement X(3851).
midpoint of X(i) and X(j) for these {i,j}: {{3, 3090}, {3523, 3526}, {3528, 3851}}.
reflection of X(i) in X(j) for these {i,j}: {{550, 3528}, {3526, 140}, {3857, 3090}}.
9 X[2] + 5 X[3], 11 X[3] + 3 X[4], 4 X[4] - 11 X[5], 12 X[2] - 5 X[5], 4 X[3] + 3 X[5], 17 X[3] - 3 X[20], 17 X[5] + 4 X[20], 17 X[4] + 11 X[20], 3 X[2] - 10 X[140], X[5] - 8 X[140], X[3] + 6 X[140], 19 X[5] - 12 X[381], 19 X[2] - 5 X[381], 19 X[3] + 9 X[381], 15 X[2] - X[382], 3 X[382] - 10 X[546], 15 X[5] - 8 X[546], 9 X[2] - 2 X[546], 15 X[140] - X[546], 5 X[3] + 2 X[546], 17 X[2] - 10 X[547], 17 X[140] - 3 X[547], X[20] + 6 X[547], 17 X[3] + 18 X[547], 13 X[3] - 6 X[548], 13 X[140] + X[548], 13 X[5] + 8 X[548], 13 X[546] + 15 X[548], 2 X[3] - 9 X[549], 4 X[140] + 3 X[549], 2 X[2] + 5 X[549], X[5] + 6 X[549], 4 X[547] + 17 X[549], 2 X[381] + 19 X[549], 10 X[20] - 17 X[550], 10 X[3] - 3 X[550], 15 X[549] - X[550], 6 X[2] + X[550], 5 X[5] + 2 X[550], 4 X[546] + 3 X[550], 2 X[382] + 5 X[550], 10 X[4] + 11 X[550], X[3] - 15 X[631], 3 X[549] - 10 X[631], 2 X[140] + 5 X[631], 3 X[5] - 10 X[632], 12 X[140] - 5 X[632], 6 X[631] + X[632], 2 X[3] + 5 X[632], 9 X[549] + 5 X[632], 13 X[632] - 6 X[1656], 13 X[631] + X[1656], 2 X[548] + 5 X[1656], 13 X[3] + 15 X[1656], 9 X[381] - 19 X[3090], 18 X[547] - 17 X[3090], 15 X[1656] - 13 X[3090], 3 X[4] - 11 X[3090], 9 X[2] - 5 X[3090], 2 X[546] - 5 X[3090], 3 X[5] - 4 X[3090], 5 X[632] - 2 X[3090], 6 X[140] - X[3090], 15 X[631] + X[3090], 9 X[549] + 2 X[3090], 3 X[550] + 10 X[3090], 6 X[548] + 13 X[3090], 3 X[20] + 17 X[3090], 9 X[3090] - 5 X[3091], 9 X[632] - 2 X[3091], 9 X[3] + 5 X[3091], 15 X[1656] - X[3146], 13 X[3090] - X[3146], 13 X[3] + X[3146], 6 X[548] + X[3146], X[20] - 17 X[3523], 2 X[548] - 13 X[3523], X[550] - 10 X[3523], X[3] - 3 X[3523], 3 X[549] - 2 X[3523], 5 X[631] - X[3523], 2 X[140] + X[3523], X[3090] + 3 X[3523], X[5] + 4 X[3523], 3 X[2] + 5 X[3523], 5 X[632] + 6 X[3523], X[4] + 11 X[3523], 5 X[1656] + 13 X[3523], 2 X[546] + 15 X[3523], 6 X[547] + 17 X[3523], 3 X[381] + 19 X[3523], 2 X[548] - 9 X[3524], 13 X[3523] - 9 X[3524], 13 X[549] - 6 X[3524], 5 X[1656] + 9 X[3524], 13 X[2] + 15 X[3524], 18 X[140] - 11 X[3525], 3 X[3090] - 11 X[3525], 3 X[3] + 11 X[3525], 9 X[3523] + 11 X[3525], 3 X[381] - 19 X[3526], 6 X[547] - 17 X[3526], 2 X[546] - 15 X[3526], 5 X[1656] - 13 X[3526], X[4] - 11 X[3526], 11 X[3525] - 9 X[3526], 5 X[632] - 6 X[3526], 3 X[2] - 5 X[3526], X[5] - 4 X[3526], X[3090] - 3 X[3526], 5 X[631] + X[3526], 3 X[549] + 2 X[3526], X[3] + 3 X[3526], X[550] + 10 X[3526], 2 X[548] + 13 X[3526], 9 X[3524] + 13 X[3526], X[20] + 17 X[3526], 5 X[20] - 17 X[3528], 10 X[548] - 13 X[3528], 5 X[3] - 3 X[3528], 15 X[549] - 2 X[3528], 5 X[3523] - X[3528], 3 X[2] + X[3528], 10 X[140] + X[3528], 5 X[3526] + X[3528], 2 X[546] + 3 X[3528], 5 X[3090] + 3 X[3528], 5 X[5] + 4 X[3528], X[382] + 5 X[3528], 5 X[4] + 11 X[3528], 15 X[381] + 19 X[3528], 9 X[550] - 2 X[3529], 15 X[3] - X[3529], 9 X[3528] - X[3529], 6 X[546] + X[3529], 15 X[3090] + X[3529], 9 X[382] + 5 X[3529], 15 X[3146] + 13 X[3529], 5 X[3] - 12 X[3530], 15 X[549] - 8 X[3530], X[550] - 8 X[3530], 5 X[3523] - 4 X[3530], X[3528] - 4 X[3530], 5 X[140] + 2 X[3530], 3 X[2] + 4 X[3530], 5 X[3526] + 4 X[3530], X[546] + 6 X[3530], 5 X[3090] + 12 X[3530], 5 X[5] + 16 X[3530], 3 X[381] - 17 X[3533], 19 X[3526] - 17 X[3533], 19 X[3523] + 17 X[3533], 5 X[1385] + 2 X[3626], 6 X[3146] - 13 X[3627], 18 X[4] - 11 X[3627], 12 X[546] - 5 X[3627], 10 X[3091] - 3 X[3627], 9 X[5] - 2 X[3627], 15 X[632] - X[3627], 6 X[3090] - X[3627], 18 X[3526] - X[3627], 6 X[3] + X[3627], 18 X[3523] + X[3627], 9 X[550] + 5 X[3627], 18 X[3528] + 5 X[3627], 2 X[3529] + 5 X[3627], 18 X[20] + 17 X[3627], 9 X[5] - 16 X[3628], 5 X[3091] - 12 X[3628], 3 X[546] - 10 X[3628], 15 X[632] - 8 X[3628], X[3627] - 8 X[3628], 3 X[3090] - 4 X[3628], 11 X[3525] - 4 X[3628], 9 X[3526] - 4 X[3628], 9 X[140] - 2 X[3628], 3 X[3] + 4 X[3628], 9 X[3523] + 4 X[3628], 9 X[3530] + 5 X[3628], 10 X[575] - 3 X[3629], 5 X[182] + 2 X[3631], 5 X[1483] + 2 X[3632], 7 X[3627] - 18 X[3832], 14 X[546] - 15 X[3832], 7 X[4] - 11 X[3832], 7 X[5] - 4 X[3832], 7 X[3090] - 3 X[3832], 14 X[140] - X[3832], 7 X[3526] - X[3832], 7 X[3523] + X[3832], 7 X[3] + 3 X[3832], 7 X[3528] + 5 X[3832], 7 X[550] + 10 X[3832], 14 X[548] + 13 X[3832], 7 X[20] + 17 X[3832], 2 X[3146] - 9 X[3845], 13 X[5] - 6 X[3845], 10 X[1656] - 3 X[3845], 13 X[549] + X[3845], 6 X[3524] + X[3845], 4 X[548] + 3 X[3845], 13 X[550] + 15 X[3845], 3 X[376] + 4 X[3850], 15 X[381] - 19 X[3851], 5 X[3627] - 18 X[3851], 5 X[4] - 11 X[3851], 5 X[3832] - 7 X[3851], X[382] - 5 X[3851], 5 X[5] - 4 X[3851], 2 X[546] - 3 X[3851], 5 X[3090] - 3 X[3851], 10 X[140] - X[3851], 5 X[3526] - X[3851], 5 X[3523] + X[3851], 4 X[3530] + X[3851], 15 X[549] + 2 X[3851], X[550] + 2 X[3851], 5 X[3] + 3 X[3851], X[3529] + 9 X[3851], 10 X[548] + 13 X[3851], 5 X[20] + 17 X[3851], 5 X[3522] + 2 X[3853], 13 X[3851] - 11 X[3855], 10 X[548] + 11 X[3855], 13 X[3528] + 11 X[3855], 18 X[381] - 19 X[3857], 2 X[3146] - 13 X[3857], 9 X[3845] - 13 X[3857], 6 X[4] - 11 X[3857], 10 X[3091] - 9 X[3857], 6 X[3832] - 7 X[3857], 18 X[2] - 5 X[3857], 4 X[546] - 5 X[3857], 6 X[3851] - 5 X[3857], X[3627] - 3 X[3857], 8 X[3628] - 3 X[3857], 3 X[5] - 2 X[3857], 12 X[140] - X[3857], 5 X[632] - X[3857], 6 X[3526] - X[3857], 2 X[3] + X[3857], 9 X[549] + X[3857], 6 X[3523] + X[3857], 3 X[550] + 5 X[3857], 6 X[3528] + 5 X[3857], 12 X[548] + 13 X[3857], 2 X[3529] + 15 X[3857], 6 X[20] + 17 X[3857], 17 X[3857] - 15 X[3858], 17 X[5] - 10 X[3858], 12 X[547] - 5 X[3858], 17 X[632] - 3 X[3858], 2 X[20] + 5 X[3858], 3 X[3543] - 10 X[3859], 3 X[3534] + 4 X[3861], X[2] - 15 X[5054], 2 X[140] - 9 X[5054], X[3526] - 9 X[5054], X[549] + 6 X[5054], 5 X[631] + 9 X[5054], X[3523] + 9 X[5054], X[3524] + 13 X[5054], 2 X[3853] - 9 X[5055], 5 X[3522] + 9 X[5055], 4 X[3861] - 11 X[5056], 3 X[3534] + 11 X[5056], X[1657] + 6 X[5066], 6 X[5066] - 13 X[5067], X[1657] + 13 X[5067], 4 X[3850] - 11 X[5070], 3 X[376] + 11 X[5070], 8 X[3856] - 15 X[5071], 18 X[547] - 11 X[5072], 17 X[3090] - 11 X[5072], 17 X[3525] - 3 X[5072], 17 X[3] + 11 X[5072], 3 X[20] + 11 X[5072], 8 X[3856] - X[5073], 15 X[5071] - X[5073], 12 X[3850] - 5 X[5076], 9 X[376] + 5 X[5076], 6 X[546] - 13 X[5079], 15 X[3090] - 13 X[5079], 9 X[3851] - 13 X[5079], 15 X[3] + 13 X[5079], 9 X[3528] + 13 X[5079], X[3529] + 13 X[5079], 2 X[3244] + 5 X[5690], 4 X[5447] + 3 X[5946], 6 X[5892] + X[6101], 6 X[3819] + X[6102], 5 X[1484] + 2 X[6154], 5 X[576] - 12 X[6329], 2 X[3636] + 5 X[6684], X[5609] + 6 X[6699], 3 X[3830] - 17 X[7486], 6 X[5901] + X[7991], 8 X[550] - 15 X[8703], 16 X[3528] - 15 X[8703], 16 X[3] - 9 X[8703], 16 X[3523] - 3 X[8703], 8 X[549] - X[8703], 4 X[5] + 3 X[8703], 16 X[3526] + 3 X[8703], 16 X[2] + 5 X[8703], 16 X[3090] + 9 X[8703], 8 X[3857] + 9 X[8703], 8 X[3845] + 13 X[8703], 16 X[3851] + 15 X[8703], 16 X[381] + 19 X[8703], 2 X[7843] - 9 X[9771], 5 X[3843] - 12 X[10109], 5 X[1656] - 12 X[10124], 13 X[3526] - 12 X[10124], X[3845] - 8 X[10124], 13 X[140] - 6 X[10124], 3 X[3524] + 4 X[10124], X[548] + 6 X[10124], 13 X[549] + 8 X[10124], 13 X[3523] + 12 X[10124], 13 X[3530] + 15 X[10124], X[3244] - 15 X[10165], X[5690] + 6 X[10165], X[6247] + 6 X[10182], X[2883] + 6 X[10193], 12 X[3636] - 5 X[10222], 6 X[6684] + X[10222], X[7575] + 6 X[10257], 2 X[7982] - 9 X[10283], 5 X[3528] - 13 X[10299], 5 X[5079] + 9 X[10299], 15 X[2] + 13 X[10299], X[382] + 13 X[10299], 5 X[3851] + 13 X[10299], X[5079] - 15 X[10303], 6 X[140] - 13 X[10303], X[3090] - 13 X[10303], 3 X[3526] - 13 X[10303], X[3] + 13 X[10303], 15 X[631] + 13 X[10303], 3 X[3523] + 13 X[10303], 4 X[10109] + 3 X[10304], 5 X[3843] + 9 X[10304], 15 X[5050] - X[11008], 9 X[10519] + 5 X[11482], 8 X[2] - 15 X[11539], 2 X[5] - 9 X[11539], 16 X[140] - 9 X[11539], 8 X[3526] - 9 X[11539], 8 X[5054] - X[11539], 4 X[549] + 3 X[11539], X[8703] + 6 X[11539], 8 X[3523] + 9 X[11539], 8 X[3524] + 13 X[11539], 19 X[140] - 12 X[11540], X[381] - 8 X[11540], 19 X[549] + 16 X[11540], 15 X[3843] - X[11541], 9 X[5650] - 2 X[11591], X[10625] - 8 X[11592], X[10263] - 8 X[11695], 11 X[546] - 18 X[11737], 5 X[4] - 12 X[11737], 11 X[3851] - 12 X[11737], 11 X[2] - 4 X[11737], 11 X[3530] + 3 X[11737], 11 X[3528] + 12 X[11737], X[3530] - 15 X[11812], 5 X[631] - 12 X[11812], X[3523] - 12 X[11812], X[549] - 8 X[11812], 3 X[5054] + 4 X[11812], X[140] + 6 X[11812], X[3526] + 12 X[11812], X[10124] + 13 X[11812], 2 X[11540] + 19 X[11812], 6 X[10691] + X[11819], 3 X[3917] + 4 X[12006], 3 X[10226] + 4 X[12010], 11 X[3] - 18 X[12100], 11 X[3523] - 6 X[12100], 11 X[549] - 4 X[12100], 11 X[140] + 3 X[12100], 2 X[11737] + 5 X[12100], X[4] + 6 X[12100], 11 X[3526] + 6 X[12100], 11 X[2] + 10 X[12100], 11 X[547] + 17 X[12100], 11 X[3090] + 18 X[12100], X[5059] + 6 X[12101], 17 X[546] - 10 X[12102], 17 X[3857] - 8 X[12102], 15 X[3858] - 8 X[12102], 17 X[3090] - 4 X[12102], 11 X[5072] - 4 X[12102], 17 X[3628] - 3 X[12102], 9 X[547] - 2 X[12102], 17 X[3] + 4 X[12102], 3 X[20] + 4 X[12102], 3 X[3529] - 10 X[12103], 9 X[3] - 2 X[12103], 6 X[3628] + X[12103], 9 X[3090] + 2 X[12103], 5 X[3091] + 2 X[12103], 3 X[3627] + 4 X[12103], 9 X[3857] + 4 X[12103], 9 X[546] + 5 X[12103], 18 X[12102] + 17 X[12103], 3 X[5499] + 4 X[12104], 6 X[5498] + X[12107], 9 X[549] - 16 X[12108], 3 X[3530] - 10 X[12108], X[3] - 8 X[12108], 15 X[631] - 8 X[12108], 3 X[3523] - 8 X[12108], 9 X[11812] - 2 X[12108], 3 X[140] + 4 X[12108], X[3628] + 6 X[12108], X[3090] + 8 X[12108], 3 X[3526] + 8 X[12108], 13 X[10303] + 8 X[12108], 5 X[632] + 16 X[12108], X[3857] + 16 X[12108], 9 X[11540] + 19 X[12108], 9 X[3845] - 16 X[12811], 13 X[3857] - 16 X[12811], 15 X[1656] - 8 X[12811], 13 X[3090] - 8 X[12811], X[3146] - 8 X[12811], 13 X[3628] - 6 X[12811], 9 X[10124] - 2 X[12811], 13 X[12108] + X[12811], 3 X[548] + 4 X[12811], 13 X[3] + 8 X[12811], 11 X[3091] - 18 X[12812], 3 X[4] - 10 X[12812], 11 X[3090] - 10 X[12812], 11 X[632] - 4 X[12812], 9 X[12100] + 5 X[12812], 11 X[3] + 10 X[12812], 4 X[11592] + 3 X[13363], X[10625] + 6 X[13363], 3 X[9143] + 4 X[13393], 6 X[10170] + X[13491], 8 X[13372] - X[14073], X[5059] - 15 X[14093], 2 X[12101] + 5 X[14093], 17 X[3851] - 9 X[14269], 17 X[2] - 3 X[14269], 10 X[547] - 3 X[14269], 5 X[20] + 9 X[14269], 17 X[3528] + 9 X[14269], 17 X[550] + 18 X[14269].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3, 546), (2, 550, 5), (2, 3523, 3528), (2, 3528, 3851), (2, 3529, 5079), (2, 3530, 550), (2, 3855, 1656), (2, 5079, 3628), (2, 10299, 382), (2, 14269, 547), (3, 140, 632), (3, 546, 550), (3, 631, 12108), (3, 632, 5), (3, 1656, 3146), (3, 3091, 12103), (3, 3146, 548), (3, 3525, 3628), (3, 3526, 3090), (3, 3628, 3627), (3, 5054, 10303), (3, 5070, 5076), (3, 5072, 20), (3, 5076, 376), (3, 5079, 3529), (3, 10303, 140), (3, 12108, 549), (5, 140, 11539), (20, 547, 3858), (20, 5072, 12102), (140, 548, 10124), (140, 549, 5), (140, 631, 549), (140, 3530, 2), (140, 3628, 3525), (140, 11812, 631), (140, 12108, 3), (376, 5070, 3850), (546, 3529, 3627), (546, 3530, 3), (546, 3628, 5079), (546, 3851, 3857), (546, 12102, 14269), (546, 12811, 3855), (546, 12812, 11737), (547, 3858, 5), (547, 12102, 5072), (548, 1656, 3845), (548, 10124, 1656), (548, 12811, 3146), (549, 550, 3530), (549, 632, 3), (549, 3845, 3524), (549, 11539, 8703), (550, 3627, 3529), (631, 5054, 140), (631, 10303, 3), (632, 3627, 3628), (1656, 3146, 12811), (1656, 3524, 548), (1656, 3845, 5), (1657, 5067, 5066), (3090, 3523, 3), (3090, 3857, 5), (3091, 12103, 3627), (3146, 3524, 3), (3146, 12811, 3845), (3522, 5055, 3853), (3524, 10124, 3845), (3525, 3529, 2), (3525, 3628, 632), (3526, 3851, 2), (3529, 5079, 546), (3534, 5056, 3861), (3627, 3628, 5), (3628, 12103, 3091), (5010, 7294, 10593), (5054, 11812, 549), (5071, 5073, 3856), (5072, 12102, 3858), (5326, 7280, 10592), (10303, 12108, 632), (11592, 13363, 10625), (14784, 14785, 3854).

Best regards,
Peter Moses.