Circumcenters - Orthologic

Let ABC be a triangle and P a point.

Denote:

Bc, Cb = the orthogonal projections of B, C on PC, PB, resp.

Oa = the circumcenter of ABcCb.
Similarly Ob, Oc.

Which is the locus of P such that ABC, OaObOc are orthologic?

H, O, G lie on the locus.

ORTHOLOGIC TRIANGLES

Let ABC be a triangle, P, Q two points, A'B'C' the padal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.
Sa = same to S point of the triangle A'AbAc.
Similary Sb,Sc.

Cases the triangle ABC, SaSbSc are orthologic.

1. Let ABC be a triangle, P a point and A'B'C' the pedal triangle of a point Q.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Ga = the centroid of A'AbAc.
Similarly Gb, Gc

ABC, GaGbGc are circumorthologic.
ie The Orthologic center (ABC, GaGbGc) = X1 lies on the circumcircle of ABC.
The Orthologic center (GaGbGc, ABC) = X2 lies on the circumcircle of GaGbGc.

2. Let ABC be a triangle, P = I = X(1), A'B'C' the pedal triangle of a point Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are Orthologic.

3. Let ABC be a triangle, P, Q two isogonal conjugate points, A'B'C' the pedal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are circumoerthologic. ie The Orthologic center (ABC, SaSbSc) = X1 lies on the circumcircle of ABC.
The Orthologic center (SaSbSc, ABC) = X2 lies on the circumcircle of SaSbSc.

EULER

X(72398) = 105TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 6*a^10 - 11*a^8*b^2 - 2*a^6*b^4 + 12*a^4*b^6 - 4*a^2*b^8 - b^10 - 11*a^8*c^2 + 38*a^6*b^2*c^2 - 21*a^4*b^4*c^2 - 9*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 21*a^4*b^2*c^4 + 26*a^2*b^4*c^4 - 2*b^6*c^4 + 12*a^4*c^6 - 9*a^2*b^2*c^6 - 2*b^4*c^6 - 4*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(72398) = 5 X[3] - 3 X[16532], 3 X[3] - X[43893], 7 X[3] - 3 X[46451], X[23] - 4 X[62087], 3 X[140] - 2 X[403], 5 X[140] - 4 X[15350], 5 X[140] - 2 X[44267], 3 X[376] + X[35452], 4 X[403] - 3 X[11558], 5 X[403] - 6 X[15350], X[403] - 3 X[34152], 5 X[403] - 3 X[44267], 4 X[468] - 7 X[61784], 5 X[546] - 8 X[5159], 7 X[546] - 8 X[63838], 3 X[547] - 4 X[10257], 7 X[547] - 4 X[47310], 25 X[548] - 4 X[37899], 9 X[548] - 4 X[37931], 7 X[548] - 4 X[47335], 19 X[548] - 4 X[47342], 3 X[549] - X[52403], 5 X[550] + X[5189], 3 X[550] - X[13619], 2 X[858] + X[62151], X[1657] + 3 X[44450], X[2070] - 3 X[8703], 5 X[2071] - X[18403], 3 X[2071] - X[37938], 9 X[2071] - X[64890], 7 X[2072] - 3 X[65087], 5 X[3522] - X[5899], 7 X[3528] - 3 X[37922], 2 X[3530] - 3 X[37948], 3 X[3534] + X[46450], X[3627] - 3 X[65085], 7 X[3853] - 6 X[65087], 3 X[5066] - 2 X[44283], 7 X[5159] - 5 X[63838], 3 X[5189] + 5 X[13619], X[5189] - 5 X[18859], 3 X[7426] - 5 X[15646], 2 X[7426] - 5 X[34200], X[7464] + 2 X[44245], X[7574] + 2 X[62136], 5 X[10096] - 6 X[16532], 3 X[10096] - 2 X[43893], 7 X[10096] - 6 X[46451], 4 X[10151] - 5 X[61940], 7 X[10257] - 3 X[47310], 5 X[11558] - 8 X[15350], X[11558] - 4 X[34152], 5 X[11558] - 4 X[44267], X[11563] - 3 X[37948], 2 X[11799] - 5 X[61790], 3 X[12100] - 2 X[44234], 3 X[12101] - 4 X[23323], X[12103] + 2 X[37950], 5 X[12812] - 2 X[62288], X[13473] - 3 X[15122], 4 X[13473] - 3 X[62026], X[13619] + 3 X[18859], 3 X[14893] - 2 X[64891], 4 X[15122] - X[62026], 2 X[15350] - 5 X[34152], 2 X[15646] - 3 X[34200], 9 X[15688] - X[37949], 3 X[15690] - 2 X[44246], X[15690] + 2 X[54995], 5 X[15712] - 3 X[37943], 9 X[16532] - 5 X[43893], 7 X[16532] - 5 X[46451], 8 X[16976] - 7 X[61821], X[18325] - 4 X[61792], 3 X[18403] - 5 X[37938], 9 X[18403] - 5 X[64890], 2 X[18572] + X[62156], X[20063] - 13 X[62105], 5 X[22248] - 2 X[62344], 3 X[25338] - 4 X[37935], 7 X[25338] - 4 X[47338], 2 X[25338] - 5 X[62064], 5 X[30745] - 2 X[62013], 5 X[34152] - X[44267], X[35001] + 5 X[62104], 3 X[35489] - 7 X[62100], 5 X[37760] - 11 X[62062], 9 X[37899] - 25 X[37931], 7 X[37899] - 25 X[47335], 19 X[37899] - 25 X[47342], 5 X[37923] - 17 X[62084], X[37924] - 7 X[62091], X[37925] - 6 X[62089], 7 X[37931] - 9 X[47335], 19 X[37931] - 9 X[47342], 2 X[37935] - 3 X[37968], 7 X[37935] - 3 X[47338], 8 X[37935] - 15 X[62064], 3 X[37938] - X[64890], 3 X[37941] - 4 X[58190], X[37944] + 4 X[41981], X[37947] - 3 X[44280], 3 X[37955] - 2 X[44264], 3 X[37955] - 5 X[46853], 3 X[37956] - 11 X[62085], 5 X[37958] - 11 X[62079], 7 X[37968] - 2 X[47338], 4 X[37968] - 5 X[62064], 9 X[41982] - 8 X[47114], 3 X[41983] - 2 X[44282], 7 X[43893] - 9 X[46451], 2 X[44214] - 3 X[61782], X[44246] + 3 X[54995], 2 X[44264] - 5 X[46853], 4 X[44452] - 5 X[61810], 8 X[44911] - 9 X[47598], 4 X[44961] - 7 X[61821], 4 X[46031] - 5 X[48154], 4 X[47090] + X[58203], 4 X[47311] + 5 X[62138], 19 X[47335] - 7 X[47342], 4 X[47336] - 7 X[55862], 8 X[47338] - 35 X[62064], 7 X[50693] + X[60466], 5 X[60455] + 7 X[62134], 3 X[60462] + 5 X[62131], 13 X[62092] - X[62290], 3 X[13363] - 2 X[13446]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72398) lies on these lines: {2, 3}, {74, 50708}, {477, 33639}, {930, 67735}, {1154, 17855}, {1291, 67797}, {1294, 13863}, {2693, 30248}, {2777, 46114}, {6799, 53934}, {13363, 13446}, {13391, 37853}, {13399, 32423}, {13445, 34153}, {14677, 43574}, {22115, 43391}, {29011, 67784}, {40111, 50434}, {53884, 67727}

X(72398) = midpoint of X(i) and X(j) for these {i,j}: {550, 18859}, {3153, 15704}, {13445, 34153}, {14677, 43574}, {16386, 37950}, {40111, 50434}
X(72398) = reflection of X(i) in X(j) for these {i,j}: {140, 34152}, {186, 33923}, {3853, 2072}, {10096, 3}, {11558, 140}, {11563, 3530}, {12103, 16386}, {25338, 37968}, {31726, 3628}, {44267, 15350}, {44961, 16976}, {47096, 22249}
X(72398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {403, 13473, 44226}, {427, 21284, 65154}, {3520, 13619, 403}, {3530, 50143, 140}, {5159, 7426, 6677}, {5189, 6636, 7426}, {11563, 37948, 3530}, {15690, 66718, 548}, {16387, 47311, 5159}

---

X(72399) = 106TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 4*a^10 - 12*a^8*b^2 + 8*a^6*b^4 + 8*a^4*b^6 - 12*a^2*b^8 + 4*b^10 - 12*a^8*c^2 + 2*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 29*a^2*b^6*c^2 - 12*b^8*c^2 + 8*a^6*c^4 - 7*a^4*b^2*c^4 - 34*a^2*b^4*c^4 + 8*b^6*c^4 + 8*a^4*c^6 + 29*a^2*b^2*c^6 + 8*b^4*c^6 - 12*a^2*c^8 - 12*b^2*c^8 + 4*c^10 : :
X(72399) = X[2] - 9 X[37943], X[2] - 3 X[44282], 25 X[2] - 9 X[44450], 7 X[2] + 9 X[46451], 17 X[2] - 9 X[65085], 2 X[5] + X[12105], X[23] + 3 X[5055], 7 X[140] + 2 X[47338], 3 X[186] + X[3830], 5 X[381] - X[10296], X[381] + 3 X[37907], 3 X[403] - X[3845], 3 X[403] + X[44265], 4 X[468] - X[18571], 3 X[468] - X[18579], 5 X[468] - 2 X[22249], 2 X[468] + X[44961], 9 X[468] - X[47031], 13 X[468] - X[47308], 11 X[468] + X[47309], 7 X[468] + X[47310], 3 X[468] + X[47332], 5 X[468] - X[47333], 7 X[468] - X[47335], 5 X[468] + X[47336], 3 X[549] - X[54995], 5 X[632] + X[62344], X[858] - 3 X[15699], 5 X[1656] - X[10989], 5 X[1656] + X[37967], 3 X[2070] + 5 X[19709], 3 X[2071] - 7 X[15701], 3 X[2072] + X[47313], 3 X[2072] - 5 X[61910], 7 X[3090] + X[37901], 3 X[3153] - 11 X[61932], 3 X[3524] + X[18325], X[3534] - 3 X[15646], 3 X[3545] - X[18572], 3 X[3545] + 5 X[37760], 2 X[3628] + X[16619], 5 X[3843] + 7 X[37957], 7 X[3851] + 5 X[37953], X[3853] + 2 X[37934], 3 X[5054] - X[37950], X[5066] + 3 X[10096], X[5066] - 6 X[37942], 2 X[5066] - 3 X[46031], 7 X[5066] - 6 X[63838], 11 X[5070] + X[37946], 5 X[5071] - X[7574], 5 X[5071] + 3 X[37909], 2 X[5159] - 3 X[47599], X[5189] - 9 X[61899], 3 X[5899] + 13 X[61901], X[7464] - 5 X[15694], X[7574] + 3 X[37909], 5 X[7575] + X[10296], X[7575] - 3 X[37907], X[8703] + 3 X[11563], 5 X[8703] - 3 X[16386], 2 X[8703] - 3 X[37968], X[8703] - 3 X[44214], X[8703] - 6 X[44900], X[10096] + 2 X[37942], 2 X[10096] + X[46031], 7 X[10096] + 2 X[63838], 2 X[10109] + X[37904], 3 X[10151] - 2 X[61997], 3 X[10257] - 4 X[11540], X[10296] + 15 X[37907], X[10297] + 2 X[44264], X[11001] + 3 X[31726], X[11558] + 2 X[16531], 3 X[11558] + X[62138], 5 X[11563] + X[16386], 2 X[11563] + X[37968], X[11563] + 2 X[44900], 3 X[11799] + X[54995], 2 X[11812] - 3 X[44452], X[12100] - 3 X[44234], 4 X[12811] - X[47339], 3 X[13619] + 5 X[62007], 3 X[14269] + 5 X[37958], 3 X[14892] + 4 X[47316], 2 X[15350] + X[37971], 6 X[15350] - X[47311], 3 X[15350] - 2 X[61896], X[15681] - 5 X[37952], X[15682] - 3 X[44283], X[15685] - 9 X[37955], 5 X[15693] - 3 X[34152], 5 X[15695] - 9 X[37941], 7 X[15703] + X[37924], 5 X[15713] + 3 X[43893], 2 X[16386] - 5 X[37968], X[16386] - 5 X[44214], X[16386] - 10 X[44900], 6 X[16531] - X[62138], 9 X[16532] - X[19710], 3 X[16532] - X[44280], X[18323] - 3 X[23046], 3 X[18403] - 7 X[41106], 3 X[18571] - 4 X[18579], 5 X[18571] - 8 X[22249], X[18571] + 2 X[44961], 9 X[18571] - 4 X[47031], 13 X[18571] - 4 X[47308], 11 X[18571] + 4 X[47309], 7 X[18571] + 4 X[47310], 3 X[18571] + 4 X[47332], 5 X[18571] - 4 X[47333], X[18571] + 4 X[47334], 7 X[18571] - 4 X[47335], 5 X[18571] + 4 X[47336], X[18572] + 5 X[37760], 5 X[18579] - 6 X[22249], 2 X[18579] + 3 X[44961], 3 X[18579] - X[47031], 13 X[18579] - 3 X[47308], 11 X[18579] + 3 X[47309], 7 X[18579] + 3 X[47310], 5 X[18579] - 3 X[47333], X[18579] + 3 X[47334], 7 X[18579] - 3 X[47335], 5 X[18579] + 3 X[47336], 3 X[18859] - 11 X[61843], 5 X[19708] + 3 X[52403], X[19710] - 3 X[44280], X[20063] + 15 X[61906], 5 X[22248] + 3 X[41987], 4 X[22249] + 5 X[44961], 18 X[22249] - 5 X[47031], 26 X[22249] - 5 X[47308], 22 X[22249] + 5 X[47309], 14 X[22249] + 5 X[47310], 6 X[22249] + 5 X[47332], 2 X[22249] + 5 X[47334], 14 X[22249] - 5 X[47335], 2 X[22249] + X[47336], 3 X[23323] - 4 X[61960], X[25338] + 2 X[68319], 5 X[30745] - 9 X[61887], 7 X[33699] - 9 X[65087], X[35001] - 9 X[61864], 4 X[35018] + X[47312], 4 X[35018] - X[47341], 3 X[35452] - 19 X[61857], 9 X[35489] + 7 X[62009], 2 X[37897] + 3 X[47478], X[37899] + 6 X[45757], X[37900] + 9 X[61909], 4 X[37911] - 3 X[47598], 9 X[37922] + 7 X[61974], 5 X[37923] + 11 X[61925], 3 X[37925] + 17 X[61893], 3 X[37931] + 2 X[62010], 6 X[37935] + X[62022], 3 X[37936] + 7 X[61920], 3 X[37938] - X[47314], 3 X[37938] - 7 X[61898], 9 X[37940] + 11 X[61950], 4 X[37942] - X[46031], 7 X[37942] - X[63838], 9 X[37943] + X[44266], 3 X[37943] - X[44282], 25 X[37943] - X[44450], 7 X[37943] + X[46451], 17 X[37943] - X[65085], 3 X[37944] - 23 X[61862], 3 X[37947] + 11 X[61908], 9 X[37948] - 13 X[61797], X[37968] - 4 X[44900], 3 X[37971] + X[47311], 3 X[37971] + 4 X[61896], 3 X[38335] + X[56369], 3 X[44246] - X[62154], X[44266] + 3 X[44282], 25 X[44266] + 9 X[44450], 7 X[44266] - 9 X[46451], 17 X[44266] + 9 X[65085], 25 X[44282] - 3 X[44450], 7 X[44282] + 3 X[46451], 17 X[44282] - 3 X[65085], 7 X[44450] + 25 X[46451], 17 X[44450] - 25 X[65085], 9 X[44961] + 2 X[47031], 13 X[44961] + 2 X[47308], 11 X[44961] - 2 X[47309], 7 X[44961] - 2 X[47310], 3 X[44961] - 2 X[47332], 5 X[44961] + 2 X[47333], 7 X[44961] + 2 X[47335], 5 X[44961] - 2 X[47336], 7 X[46031] - 4 X[63838], 3 X[46450] - 19 X[61913], 17 X[46451] + 7 X[65085], 13 X[47031] - 9 X[47308], 11 X[47031] + 9 X[47309], 7 X[47031] + 9 X[47310], X[47031] + 3 X[47332], 5 X[47031] - 9 X[47333], X[47031] + 9 X[47334], 7 X[47031] - 9 X[47335], 5 X[47031] + 9 X[47336], 3 X[47096] + 7 X[61851], 11 X[47308] + 13 X[47309], 7 X[47308] + 13 X[47310], 3 X[47308] + 13 X[47332], 5 X[47308] - 13 X[47333], X[47308] + 13 X[47334], 7 X[47308] - 13 X[47335], 5 X[47308] + 13 X[47336], 7 X[47309] - 11 X[47310], 3 X[47309] - 11 X[47332], 5 X[47309] + 11 X[47333], X[47309] - 11 X[47334], 7 X[47309] + 11 X[47335], 5 X[47309] - 11 X[47336], 3 X[47310] - 7 X[47332], 5 X[47310] + 7 X[47333], X[47310] - 7 X[47334], 5 X[47310] - 7 X[47336], X[47311] - 4 X[61896], X[47313] + 5 X[61910], X[47314] - 7 X[61898], 5 X[47332] + 3 X[47333], X[47332] - 3 X[47334], 7 X[47332] + 3 X[47335], 5 X[47332] - 3 X[47336], X[47333] + 5 X[47334], 7 X[47333] - 5 X[47335], 7 X[47334] + X[47335], 5 X[47334] - X[47336], 5 X[47335] + 7 X[47336], X[47340] + 4 X[67236], X[47342] + 4 X[61922], 7 X[55856] - X[62332], 3 X[57584] - 5 X[61998], 5 X[60455] - 21 X[61897], 15 X[61882] + X[62290], 7 X[62000] - 3 X[64890], X[62043] - 3 X[64891], X[110] + 3 X[15362], 3 X[5215] - X[38611], X[9158] + 3 X[57305], X[11179] - 5 X[47453], X[11801] + 2 X[15448], 3 X[14643] + X[15360], 2 X[15088] + X[32237], X[20423] + 3 X[47450], X[21850] + 5 X[47452], X[34315] + 3 X[59403], X[34316] + 3 X[59404], 3 X[47455] - X[50979], X[47471] + 3 X[47562], X[50955] + 3 X[52238]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72399) lies on these lines: {2, 3}, {110, 15362}, {113, 15361}, {524, 10272}, {952, 47495}, {3564, 47544}, {5215, 38611}, {5844, 47488}, {9158, 57305}, {11178, 32217}, {11179, 47453}, {11645, 20304}, {11649, 13364}, {11801, 15448}, {12900, 19924}, {14643, 15360}, {15088, 32237}, {16328, 18487}, {20423, 47450}, {21850, 47452}, {32423, 35266}, {32515, 46986}, {34315, 59403}, {34316, 59404}, {34380, 47473}, {43291, 47169}, {43656, 53950}, {44204, 47219}, {44569, 46817}, {45969, 61606}, {47455, 50979}, {47471, 47562}, {47556, 47581}, {50955, 52238}, {61572, 62508}, {61619, 63124}

X(72399) = midpoint of X(i) and X(j) for these {i,j}: {2, 44266}, {5, 7426}, {113, 15361}, {376, 44267}, {381, 7575}, {468, 47334}, {547, 25338}, {549, 11799}, {3845, 44265}, {10295, 15687}, {10989, 37967}, {11178, 32217}, {11563, 44214}, {11737, 44264}, {15686, 62288}, {16619, 47097}, {18579, 47332}, {44204, 47219}, {44569, 46817}, {47310, 47335}, {47312, 47341}, {47333, 47336}, {47556, 47581}
X(72399) = reflection of X(i) in X(j) for these {i,j}: {547, 68319}, {10297, 11737}, {12105, 7426}, {14893, 37984}, {15122, 10124}, {37968, 44214}, {44214, 44900}, {44961, 47334}, {47097, 3628}, {47333, 22249}, {62139, 66595}
X(72399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {381, 37907, 7575}, {403, 44265, 3845}, {403, 66725, 37984}, {468, 44961, 18571}, {468, 47332, 18579}, {468, 47336, 22249}, {5071, 37909, 7574}, {10096, 37942, 46031}, {10096, 44233, 25338}, {10109, 66529, 5066}, {10296, 10298, 16386}, {11563, 44900, 37968}, {13626, 13627, 381}, {14002, 37907, 7426}, {18579, 47334, 47332}, {25338, 44234, 25337}, {34330, 62961, 14893}, {44233, 68319, 46031}, {44266, 44282, 2}, {57322, 57323, 61924}

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TWO PROBLEMS CONNECTED

Francisco Javier García Capitán, Two problems [by Antreas Hatzipolakis] connected.

FGJC
geogebra file

z

X(72407) = X(13)X(42788)∩X(14)X(1506)

Barycentrics    8 a^16 - 84 a^14 b^2 + 375 a^12 b^4 - 908 a^10 b^6 + 1288 a^8 b^8 - 1086 a^6 b^10 + 528 a^4 b^12 - 136 a^2 b^14 + 15 b^16 - 84 a^14 c^2 + 594 a^12 b^2 c^2 - 1492 a^10 b^4 c^2 + 1258 a^8 b^6 c^2 + 552 a^6 b^8 c^2 - 1476 a^4 b^10 c^2 + 766 a^2 b^12 c^2 - 130 b^14 c^2 + 375 a^12 c^4 - 1492 a^10 b^2 c^4 + 1076 a^8 b^4 c^4 + 1002 a^6 b^6 c^4 + 391 a^4 b^8 c^4 - 1492 a^2 b^10 c^4 + 487 b^12 c^4 - 908 a^10 c^6 + 1258 a^8 b^2 c^6 + 1002 a^6 b^4 c^6 + 1082 a^4 b^6 c^6 + 862 a^2 b^8 c^6 - 1018 b^10 c^6 + 1288 a^8 c^8 + 552 a^6 b^2 c^8 + 391 a^4 b^4 c^8 + 862 a^2 b^6 c^8 + 1292 b^8 c^8 - 1086 a^6 c^10 - 1476 a^4 b^2 c^10 - 1492 a^2 b^4 c^10 - 1018 b^6 c^10 + 528 a^4 c^12 + 766 a^2 b^2 c^12 + 487 b^4 c^12 - 136 a^2 c^14 - 130 b^2 c^14 + 15 c^16 - 4 a^14 T + 28 a^12 b^2 T - 86 a^10 b^4 T + 138 a^8 b^6 T - 134 a^6 b^8 T + 78 a^4 b^10 T - 22 a^2 b^12 T + 2 b^14 T + 28 a^12 c^2 T - 112 a^10 b^2 c^2 T + 142 a^8 b^4 c^2 T - 48 a^6 b^6 c^2 T - 58 a^4 b^8 c^2 T + 56 a^2 b^10 c^2 T - 6 b^12 c^2 T - 86 a^10 c^4 T + 142 a^8 b^2 c^4 T - 112 a^6 b^4 c^4 T + 56 a^4 b^6 c^4 T - 84 a^2 b^8 c^4 T + 8 b^10 c^4 T + 138 a^8 c^6 T - 48 a^6 b^2 c^6 T + 56 a^4 b^4 c^6 T + 96 a^2 b^6 c^6 T - 4 b^8 c^6 T - 134 a^6 c^8 T - 58 a^4 b^2 c^8 T - 84 a^2 b^4 c^8 T - 4 b^6 c^8 T + 78 a^4 c^10 T + 56 a^2 b^2 c^10 T + 8 b^4 c^10 T - 22 a^2 c^12 T - 6 b^2 c^12 T + 2 c^14 T : : where T = Sqrt[3] S

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9479.

X(72407) lies on these lines: {5, 11602}, {13, 42788}, {14, 1506}, {17, 5615}, {18, 59403}, {76, 16966}, {83, 6671}, {5487, 35689}, {6115, 11606}, {10187, 25555}, {11122, 37832}, {11272, 43539}, {12817, 52649}, {16242, 62877}, {16964, 54861}

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X(72408) = X(13)X(1506)∩X(14)X(42788)

Barycentrics    8 a^16-84 a^14 b^2+375 a^12 b^4-908 a^10 b^6+1288 a^8 b^8-1086 a^6 b^10+528 a^4 b^12-136 a^2 b^14+15 b^16-84 a^14 c^2+594 a^12 b^2 c^2-1492 a^10 b^4 c^2+1258 a^8 b^6 c^2+552 a^6 b^8 c^2-1476 a^4 b^10 c^2+766 a^2 b^12 c^2-130 b^14 c^2+375 a^12 c^4-1492 a^10 b^2 c^4+1076 a^8 b^4 c^4+1002 a^6 b^6 c^4+391 a^4 b^8 c^4-1492 a^2 b^10 c^4+487 b^12 c^4-908 a^10 c^6+1258 a^8 b^2 c^6+1002 a^6 b^4 c^6+1082 a^4 b^6 c^6+862 a^2 b^8 c^6-1018 b^10 c^6+1288 a^8 c^8+552 a^6 b^2 c^8+391 a^4 b^4 c^8+862 a^2 b^6 c^8+1292 b^8 c^8-1086 a^6 c^10-1476 a^4 b^2 c^10-1492 a^2 b^4 c^10-1018 b^6 c^10+528 a^4 c^12+766 a^2 b^2 c^12+487 b^4 c^12-136 a^2 c^14-130 b^2 c^14+15 c^16+4 a^14 T-28 a^12 b^2 T+86 a^10 b^4 T-138 a^8 b^6 T+134 a^6 b^8 T-78 a^4 b^10 T+22 a^2 b^12 T-2 b^14 T-28 a^12 c^2 T+112 a^10 b^2 c^2 T-142 a^8 b^4 c^2 T+48 a^6 b^6 c^2 T+58 a^4 b^8 c^2 T-56 a^2 b^10 c^2 T+6 b^12 c^2 T+86 a^10 c^4 T-142 a^8 b^2 c^4 T+112 a^6 b^4 c^4 T-56 a^4 b^6 c^4 T+84 a^2 b^8 c^4 T-8 b^10 c^4 T-138 a^8 c^6 T+48 a^6 b^2 c^6 T-56 a^4 b^4 c^6 T-96 a^2 b^6 c^6 T+4 b^8 c^6 T+134 a^6 c^8 T+58 a^4 b^2 c^8 T+84 a^2 b^4 c^8 T+4 b^6 c^8 T-78 a^4 c^10 T-56 a^2 b^2 c^10 T-8 b^4 c^10 T+22 a^2 c^12 T+6 b^2 c^12 T-2 c^14 T : : where T = Sqrt[3] S

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9479.

X(72408) lies on these lines: {5, 11603}, {13, 1506}, {14, 42788}, {17, 59404}, {18, 5611}, {76, 16967}, {83, 6672}, {5488, 35688}, {6114, 11606}, {10188, 25555}, {11121, 37835}, {11272, 43538}, {12816, 44289}, {16241, 62876}, {16965, 54860}

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ETC

X(5459)
Let ABC be a triangle, let A', B', C' be the midpoints of BC, CA, AB. Let L_a be the perpendicular through A' to the line X(618)A'. Define L_b, L_c cyclically. Then X(5459) is the center of the equilateral triangle A''B''C'' bounded by L_a, L_b, L_c. The circumcircle of A''B''C'' passes through X(14081) and X(32552) and has squared radius (3 sqrt(3) S^3 + 9 S^2 SW + 3 sqrt(3) S SW^2 + SW^3)/(9 (3 S^2 + 2 sqrt(3) S SW + SW^2)). This circle (A''B''C'') is here named 1st Suppa circle. The 2nd Suppa circle is defined at X(5460)
(Euclid 8675, August 28, 2025)

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X(5960)
Let ABC be a triangle, let A', B', C' be the midpoints of BC, CA, AB. Let L_a be the perpendicular through A' to the line X(619)A'. Define L_b, L_c cyclically. Then X(5460) is the center of the equilateral triangle A''B''C'' bounded by L_a, L_b, L_c. The circumcircle of A''B''C'' passes through X(14082) and X(32553)and has squared radius (-3 sqrt(3) S^3 + 9 S^2 SW - 3 sqrt(3) S SW^2 + SW^3)/(9 (3 S^2 - 2 sqrt(3) S SW + SW^2)). The circle (A''B''C'') is here named 2nd Suppa circle. The 1st Suppa circle is defined at X(5459)
(Euclid 8675, August 28, 2025)

Another relationship between Napoleon cubic and Neuberg cubic

Another relationship between Napoleon cubic K005 and Neuberg cubic K001
The world of Triangle Geometry is very intrincate. There are many paths that lead to the same place.

In this case a problem from proposed by Benjamin L. Warren at Euclid 8052 and later expanded by Antreas Hatzipolakis at Euclid 8057 lead to a relationship between these two cubics.

Another relationship between Napoleon cubic and Neuberg cubic

Francisco Javier García Capitán