ETC1

X(73916) = X(4)X(5609)∩X(3830)X(13530)

Barycentrics    8*a^16-32*a^14*b^2+56*a^12*b^4-64*a^10*b^6+40*a^8*b^8+32*a^6*b^10-88*a^4*b^12+64*a^2*b^14-16*b^16-32*a^14*c^2+82*a^12*b^2*c^2-75*a^10*b^4*c^2+25*a^8*b^6*c^2-65*a^6*b^8*c^2+237*a^4*b^10*c^2-272*a^2*b^12*c^2+100*b^14*c^2+56*a^12*c^4-75*a^10*b^2*c^4+60*a^8*b^4*c^4+8*a^6*b^6*c^4-201*a^4*b^8*c^4+432*a^2*b^10*c^4-280*b^12*c^4-64*a^10*c^6+25*a^8*b^2*c^6+8*a^6*b^4*c^6+104*a^4*b^6*c^6-224*a^2*b^8*c^6+476*b^10*c^6+40*a^8*c^8-65*a^6*b^2*c^8-201*a^4*b^4*c^8-224*a^2*b^6*c^8-560*b^8*c^8+32*a^6*c^10+237*a^4*b^2*c^10+432*a^2*b^4*c^10+476*b^6*c^10-88*a^4*c^12-272*a^2*b^2*c^12-280*b^4*c^12+64*a^2*c^14+100*b^2*c^14-16*c^16 : :
X(73916) = X[4]+X[52173], X[3830]+X[13530], 3*X[5055]-X[53693]

Marian Cucoanes and Ercole Suppa, euclid 10307.

X(73916) lies on these lines: {4, 5609}, {3830, 13530}, {5055, 53693}, {44266, 67872}

X(73916) = midpoint of X(i) and X(j) for these {i,j}: {4, 52173}, {3830, 13530}
X(73916) = center of the orthopolar conic of X(52173)

---

X(73917) = X(4)X(195)∩X(5)X(930)

Barycentrics    2*a^16-9*a^14*b^2+17*a^12*b^4-17*a^10*b^6+5*a^8*b^8+13*a^6*b^10-21*a^4*b^12+13*a^2*b^14-3*b^16-9*a^14*c^2+26*a^12*b^2*c^2-29*a^10*b^4*c^2+18*a^8*b^6*c^2-25*a^6*b^8*c^2+56*a^4*b^10*c^2-57*a^2*b^12*c^2+20*b^14*c^2+17*a^12*c^4-29*a^10*b^2*c^4+20*a^8*b^4*c^4+3*a^6*b^6*c^4-44*a^4*b^8*c^4+93*a^2*b^10*c^4-60*b^12*c^4-17*a^10*c^6+18*a^8*b^2*c^6+3*a^6*b^4*c^6+18*a^4*b^6*c^6-49*a^2*b^8*c^6+108*b^10*c^6+5*a^8*c^8-25*a^6*b^2*c^8-44*a^4*b^4*c^8-49*a^2*b^6*c^8-130*b^8*c^8+13*a^6*c^10+56*a^4*b^2*c^10+93*a^2*b^4*c^10+108*b^6*c^10-21*a^4*c^12-57*a^2*b^2*c^12-60*b^4*c^12+13*a^2*c^14+20*b^2*c^14-3*c^16 : :
X(73917) = 3*X[2]-4*X[25339], X[3]-3*X[25147], X[4]+X[1263], 3*X[4]+X[38587], 3*X[1263]-X[38587], 3*X[5]-X[930], 2*X[5]-X[6592], 5*X[5]-3*X[57316], 2*X[930]-3*X[6592], 5*X[930]-9*X[57316], 5*X[6592]-6*X[57316], 2*X[137]-X[12026], 3*X[137]-X[38618], 5*X[137]-X[63409], 3*X[12026]-2*X[38618], 5*X[12026]-2*X[63409], 5*X[38618]-3*X[63409], X[128]-2*X[3850], 3*X[140]-4*X[58432], X[140]-2*X[61594], 2*X[58432]-3*X[61594], 3*X[381]+X[11671], 3*X[381]-X[14072], X[11671]+X[14072], X[382]+3*X[47065], 3*X[547]-2*X[13372], X[548]-2*X[34837], X[550]+X[44976], X[550]-3*X[57324], X[44976]+3*X[57324], 5*X[632]-3*X[38706], X[1141]+X[3627], 5*X[3091]-X[13512], 5*X[3091]-3*X[23237], X[13512]-3*X[23237], 2*X[3530]-X[63412], 2*X[3628]-3*X[23516], 2*X[3628]-X[38615], 3*X[23516]-X[38615], 7*X[3832]-X[23238], 5*X[3843]-X[67091], 3*X[3845]-X[31656], 7*X[3857]-X[38681], 5*X[3858]-X[14073], 3*X[5066]-2*X[61587], X[6343]-3*X[61715], 5*X[12812]-4*X[58429], 3*X[13451]-2*X[68069], 3*X[15687]-X[44981], X[15704]-3*X[38710]

Marian Cucoanes and Ercole Suppa, euclid 10307.

X(73917) lies on the circumconic {{A,B,C,X(25148),X(68638)}} and these lines: {2, 25339}, {3, 25147}, {4, 195}, {5, 930}, {30, 137}, {128, 3850}, {140, 58432}, {381, 11671}, {382, 47065}, {546, 25150}, {547, 13372}, {548, 34837}, {550, 44976}, {632, 38706}, {1141, 3627}, {1154, 24306}, {3091, 13512}, {3530, 63412}, {3574, 20030}, {3583, 14101}, {3628, 23516}, {3832, 23238}, {3843, 67091}, {3845, 31656}, {3857, 38681}, {3858, 14073}, {5066, 61587}, {5899, 14652}, {6343, 61715}, {8254, 10285}, {11801, 45147}, {12812, 58429}, {13451, 68069}, {14143, 68467}, {15367, 61750}, {15687, 44981}, {15704, 38710}, {18378, 34418}, {20414, 22051}, {24144, 27423}, {38640, 55857}, {38683, 61988}, {45258, 61548}, {61504, 72664}

X(73917) = midpoint of X(i) and X(j) for these {i,j}: {4, 1263}, {550, 44976}, {1141, 3627}, {11671, 14072}
X(73917) = reflection of X(i) in X(j) for these {i,j}: {128, 3850}, {140, 61594}, {548, 34837}, {6592, 5}, {12026, 137}, {27423, 30531}, {31675, 22051}, {38615, 3628}, {61504, 72664}, {61548, 45258}, {63412, 3530}
X(73917) = center of circle {X(3448), X(11671), X(14072)}
X(73917) = center of the orthopolar conic of X(1263)
X(73917) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {11801, 4, 195}, {4, 11801, 68330}
X(73917) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1263, 32423}, {381, 11671, 14072}, {3091, 13512, 23237}, {23516, 38615, 3628}, {44976, 57324, 550}

---

X(73918) = X(4)X(1511)∩X(30)X(20480)

Barycentrics    (3*a^4-a^2*b^2-2*b^4-a^2*c^2+4*b^2*c^2-2*c^4)*(2*a^6-2*a^4*b^2-2*a^2*b^4+2*b^6-3*a^4*c^2+5*a^2*b^2*c^2-3*b^4*c^2+c^6)*(2*a^6-3*a^4*b^2+b^6-2*a^4*c^2+5*a^2*b^2*c^2-2*a^2*c^4-3*b^2*c^4+2*c^6) : :
X(73918) = 2*X[550]-X[67739]

Marian Cucoanes and Ercole Suppa, euclid 10307.

X(73918) lies on the circumconic with center X(550), the cubics K025, K446 and these lines: {4, 1511}, {30, 20480}, {550, 67739}, {11589, 14993}, {13481, 38730}, {34150, 37968}

X(73918) = reflection of X(67739) in X(550)
X(73918) = antigonal conjugate of X(382)
X(73918) = symgonal image of X(550)
X(73918) = X(56063)-reciprocal conjugate of X(57823)
X(73918) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(3),X(38942)}, {A,B,C,X(4),X(382)}, {A,B,C,X(1511),X(11589)}, {A,B,C,X(11270),X(44748)}}
X(73918) = barycentric product X(382)*X(56063)
X(73918) = barycentric quotient X(56063)/X(57823)

---

X(73919) = X(3)X(20480)∩X(4)X(1511)

Barycentrics    -8*a^16+24*a^14*b^2-8*a^12*b^4-40*a^10*b^6+40*a^8*b^8+8*a^6*b^10-24*a^4*b^12+8*a^2*b^14+24*a^14*c^2-98*a^12*b^2*c^2+107*a^10*b^4*c^2+27*a^8*b^6*c^2-107*a^6*b^8*c^2+55*a^4*b^10*c^2-12*a^2*b^12*c^2+4*b^14*c^2-8*a^12*c^4+107*a^10*b^2*c^4-212*a^8*b^4*c^4+108*a^6*b^6*c^4+41*a^4*b^8*c^4-12*a^2*b^10*c^4-24*b^12*c^4-40*a^10*c^6+27*a^8*b^2*c^6+108*a^6*b^4*c^6-144*a^4*b^6*c^6+16*a^2*b^8*c^6+60*b^10*c^6+40*a^8*c^8-107*a^6*b^2*c^8+41*a^4*b^4*c^8+16*a^2*b^6*c^8-80*b^8*c^8+8*a^6*c^10+55*a^4*b^2*c^10-12*a^2*b^4*c^10+60*b^6*c^10-24*a^4*c^12-12*a^2*b^2*c^12-24*b^4*c^12+8*a^2*c^14+4*b^2*c^14 : :
X(73918) = X[3]+X[20480], 3*X[3]-X[67739], 3*X[20480]+X[67739]

Marian Cucoanes and Ercole Suppa, euclid 10307.

X(73919) lies on these lines: {3, 20480}, {4, 1511}, {37968, 38609}

X(73919) = midpoint of X(3) and X(20480)

---

X(73920) = X(49)X(18349)∩X(93)X(18350)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 2*a^6*c^2 + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 2*b^6*c^2 + a^4*c^4 - a^2*b^2*c^4 + b^4*c^4)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 4*a^6*c^2 + 2*a^4*b^2*c^2 - a^2*b^4*c^2 + 6*a^4*c^4 + 2*a^2*b^2*c^4 + b^4*c^4 - 4*a^2*c^6 - 2*b^2*c^6 + c^8) : :

Antreas Hatzipolakis and Peter Moses, euclid 10308.

X(73920) lies on these lines: {49, 18349}, {93, 18350}, {110, 18351}, {186, 6243}, {63734, 66883}

X(73920) = X(i)-isoconjugate of X(j) for these (i,j): {54, 18352}, {2167, 18353}
X(73920) = X(i)-Dao conjugate of X(j) for these (i,j): {6663, 565}, {40588, 18353} .
X(73920) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 18353}, {1953, 18352}, {36412, 565}

---

ETC

X(73907) = X(4)X(1854)∩X(46)X(80)

Barycentrics    (2*a^4-a^3*b-a^2*b^2+a*b^3-b^4-a^3*c+2*a^2*b*c-a*b^2*c-a^2*c^2-a*b*c^2+2*b^2*c^2+a*c^3-c^4)*(2*a^6-a^5*b-a^4*b^2+2*a^3*b^3-4*a^2*b^4-a*b^5+3*b^6-a^5*c+2*a^4*b*c-2*a^3*b^2*c+3*a*b^4*c-2*b^5*c-a^4*c^2-2*a^3*b*c^2+8*a^2*b^2*c^2-2*a*b^3*c^2-3*b^4*c^2+2*a^3*c^3-2*a*b^2*c^3+4*b^3*c^3-4*a^2*c^4+3*a*b*c^4-3*b^2*c^4-a*c^5-2*b*c^5+3*c^6) : :
X(73907) = 2*X[4]-X[38357], X[102]-2*X[60758], 2*X[117]-X[38554], 4*X[117]-3*X[51408], 2*X[1535]-X[51424], 2*X[1542]-X[51361], X[10017]-2*X[72517], 2*X[38554]-3*X[51408], X[2968]-2*X[67226], X[10726]+X[18339], 2*X[15252]-X[67464]

Let A'B'C' be the orthic triangle, P a point and Pa, Pb, Pc the P-points of AB'C', A'BC', A'B'C, resp. The perpendiculars from Pa, Pb, Pc to BC, CA, AB, resp. are concurrent. The locus of the point of concurrence, which always is the crosssum of X(3) and P, is a conic passing through X(i) for i = 125, 1146, 1562, 13202, 38357, 38388, 38389, 57424, 57430, 57445, 72568, 73909, 73910, 73911, , here named Hatzipolakis - García Capitán conic.

Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 10286.

X(73907) lies on the Hatzipolakis - García Capitán conic and these lines: {4, 1854}, {19, 1146}, {46, 80}, {102, 60758}, {117, 515}, {125, 407}, {208, 1837}, {900, 42755}, {952, 10696}, {1503, 41499}, {1562, 1901}, {1718, 63988}, {1783, 5776}, {1827, 38388}, {1828, 12688}, {1844, 7686}, {1845, 6001}, {1846, 57445}, {2968, 64507}, {10726, 18339}, {15252, 67464}, {18391, 67169}, {18480, 19904}

X(73907) = midpoint of X(10726) and X(18339)
X(73907) = reflection of X(i) in X(j) for these {i,j}: {102, 60758}, {2968, 67226}, {10017, 72517}, {38357, 4}, {38554, 117}, {51361, 1542}, {51424, 1535}, {67464, 15252}
X(73907) = crosspoint and X(4) and X(515)
X(73907) = crosssum of X(3) and X(102)
X(73907) = orthopole of trilinear polar of X(52780)
X(73907) = Zosma transform of X(36121)
X(73907) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(4),X(38554)}, {A,B,C,X(80),X(51375)}, {A,B,C,X(84),X(11700)}, {A,B,C,X(36121),X(46974)}, {A,B,C,X(36127),X(66957)}}
X(73907) = center of circle {X(i),X(j),X(k)} for these {i,j,k}: {10726, 10771, 18339}
X(73907) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {42755, 10696, 10740}, {10696, 42755, 52836}
X(73907) = pole of line {39471, 53152} with respect to the polar circle
X(73907) = pole of tripolar of X(515) with respect to the orthic inconic
X(73907) = pole of line {125, 2968} with respect to the orthoptic circle of Jerabek hyperbola
X(73907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {117, 38554, 51408}

---

X(73908) = X(4)X(1146)∩X(33)X(1836)

Barycentrics    (2*a^3-a^2*b-b^3-a^2*c+b^2*c+b*c^2-c^3)*(2*a^5-a^4*b-2*a^2*b^3-2*a*b^4+3*b^5-a^4*c+2*a^2*b^2*c-b^4*c+2*a^2*b*c^2+4*a*b^2*c^2-2*b^3*c^2-2*a^2*c^3-2*b^2*c^3-2*a*c^4-b*c^4+3*c^5) : :
X(73908) = 2*X[4]-X[1146], X[20]-2*X[17044], 2*X[103]-3*X[61673], X[103]-2*X[68552], 3*X[61673]-4*X[68552], 4*X[118]-3*X[51406], 2*X[118]-X[65745], 2*X[118]-3*X[72418], X[910]-2*X[1541], 2*X[1530]-X[17747], 3*X[51406]-2*X[65745], X[51406]-2*X[72418], X[65745]-3*X[72418], X[664]+X[3146], X[1121]-3*X[50687], X[1565]-2*X[31851], 3*X[1699]-2*X[62674], 5*X[3091]-4*X[40483], 7*X[3832]-5*X[31640], 3*X[9812]-X[14942], X[10727]+X[67568], 5*X[17578]-X[39351], X[33521]-2*X[58898], X[39357]+3*X[62032], 2*X[65808]-X[67721]

Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 10286.

X(73908) lies the circunconic {{A,B,C,X(4),X(65745)}}, the Hatzipolakis - García Capitán conic and these lines: {4, 1146}, {20, 17044}, {30, 35110}, {33, 1836}, {65, 38388}, {103, 61673}, {118, 516}, {125, 430}, {152, 5845}, {223, 9580}, {528, 1750}, {664, 3146}, {952, 10725}, {1086, 60017}, {1121, 50687}, {1360, 69805}, {1503, 52468}, {1562, 1834}, {1565, 31851}, {1699, 62674}, {1824, 38389}, {1830, 1864}, {2785, 39838}, {2901, 22035}, {2910, 41869}, {3058, 20277}, {3091, 40483}, {3543, 64462}, {3832, 31640}, {4872, 70607}, {6001, 71374}, {6366, 52836}, {9579, 62793}, {9812, 14942}, {10727, 67568}, {17578, 39351}, {18328, 53804}, {33521, 58898}, {36990, 64130}, {39357, 62032}, {65808, 67721}

X(73908) = midpoint of X(i) and X(j) for these {i,j}: {664, 3146}, {10727, 67568}
X(73908) = reflection of X(i) in X(j) for these {i,j}: {20, 17044}, {103, 68552}, {910, 1541}, {1146, 4}, {1565, 31851}, {17747, 1530}, {33521, 58898}, {51406, 72418}, {65745, 118}, {67721, 65808}
X(73908) = reflection of X(i) in X(j)X(k) for these {i,j,k}}: {1566, 4, 514}
X(73908) = crosspoint of X(4) and X(516)
X(73908) = crossum of X(3) and X(103)
X(73908) = Zosma transform of X(36122)
X(73908) = orthopole of trilinear polar of X(52781)
X(73908) = pole of line {39470, 53150} with respect to the polar circle
X(73908) = pole of line {1886, 69787} with respect to the Kiepert hyperbola
X(73908) = pole of tripolar of X(516) with respect to the orthic inconic
X(73908) = pole of line {125, 1565} with respect to the orthoptic circle of Jerabek hyperbola
X(73908) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {103, 68552, 61673}, {118, 65745, 51406}, {65745, 72418, 118}

---

X(73909) = X(4)X(151)∩X(40)X(13724)

Barycentrics    a*(a^2*b-b^3+a^2*c-2*a*b*c+b^2*c+b*c^2-c^3)*(a^5*b-2*a^3*b^3+a*b^5+a^5*c-2*a^4*b*c+2*a^3*b^2*c-3*a*b^4*c+2*b^5*c+2*a^3*b*c^2+2*a*b^3*c^2-2*a^3*c^3+2*a*b^2*c^3-4*b^3*c^3-3*a*b*c^4+a*c^5+2*b*c^5) : :
X(73909) = 2*X[4]-X[38389], 2*X[104]-3*X[61674], X[104]-3*X[61731], X[61674]-2*X[61731], 4*X[119]-3*X[61672], 2*X[119]-X[65743], 2*X[119]-3*X[72417], 2*X[1512]-X[51377], X[3259]-2*X[72518], 3*X[61672]-2*X[65743], X[61672]-2*X[72417], X[65743]-3*X[72417], X[3937]-2*X[31849], 2*X[3035]-X[67420], 2*X[6713]-3*X[67216], 2*X[12019]-X[34462], X[31847]-2*X[67864], 3*X[34583]-2*X[38759], 2*X[38390]-3*X[59390], 3*X[38693]-4*X[64489], X[38761]-2*X[67414], 5*X[64008]-3*X[67634], X[67494]-2*X[68548]

Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 10286.

X(73909) lies the circunconic {{A,B,C,X(4),X(65743)}}, the Hatzipolakis - García Capitán conic and these lines: {4, 151}, {40, 13724}, {65, 1830}, {80, 2807}, {102, 52242}, {104, 61674}, {117, 867}, {119, 517}, {125, 429}, {153, 2810}, {185, 1837}, {355, 15030}, {513, 52836}, {957, 8166}, {962, 2899}, {1146, 1826}, {1361, 35015}, {1562, 53417}, {1863, 5185}, {2197, 17452}, {2779, 6246}, {2800, 22321}, {2808, 9803}, {2815, 3762}, {2823, 12736}, {2829, 3937}, {2841, 34789}, {3035, 67420}, {3040, 24410}, {3753, 25019}, {5086, 5907}, {5151, 6001}, {6256, 23154}, {6713, 67216}, {10724, 29349}, {12019, 34462}, {14127, 38607}, {17516, 63435}, {21044, 34457}, {21664, 42759}, {22306, 41507}, {22799, 61638}, {31847, 67864}, {34583, 38759}, {37437, 67968}, {38390, 59390}, {38693, 64489}, {38761, 67414}, {45022, 53548}, {53530, 72581}, {64008, 67634}, {67494, 68548}

X(73909) = reflection of X(i) in X(j) for these {i,j}: {3259, 72518}, {3937, 31849}, {31847, 67864}, {34462, 12019}, {38389, 4}, {38761, 67414}, {51377, 1512}, {61672, 72417}, {61674, 61731}, {65743, 119}, {67420, 3035}, {67494, 68548}
X(73909) = crosspoint of X(4) and X(517)
X(73909) = crosssum of X(3) and X(104)
X(73909) = perspector of the circumconic through X(2397)and X(26011)
X(73909) = orthopole of tripolar of X(16082)
X(73909) = Zosma transform of X(36123)
X(73909) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {52836, 119, 908}
X(73909) = pole of line {8677, 43933} with respect to the polar circle
X(73909) = pole of line {1877, 12138} with respect to the Feuerbach hyperbola
X(73909) = pole of tripolar of X(517) with respect to the orthic inconic
X(73909) = pole of tripolar of X(15420) with respect to the orthoptic circle of Jerabek hyperbola
X(73909) = barycentric product X(517)*X(26011)
X(73909) = barycentric quotient X(26011)/X(18816)
X(73909) = trilinear product X(2183)*X(26011)
X(73909) = trilinear quotient X(26011)/X(34234)
X(73909) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {119, 65743, 61672}, {65743, 72417, 119}

---

A CYCLOLOGIC THEOREM RELATED TO EXCENTRAL TRIANGLE.

[APH]

Excentral version

Let ABC be a triangle, IaIbIc the excentral triangle and P a point..

Denote

Pa, Pb, Pc = same to P points of IaBC, IbCA, IcAB, resp.

ABC, PaPbPc are circumcyclologic

Cyclologic center (ABC, PaPbPc) = Q = ? (on the circumcircle of ABC)
Cyclologic center (PaPbPc, ABC) = Q* = ? (on the circumcircle of PaPbPc)

[Ercole Suppa]

1. P on the Euler Line:

Q = X(100)
Locus of Q* as P moves on the Euler line: K086 2. P on the Brocard axis:

Q = X(101)

[Bernard Gibert]

If P is on a line through X(3) and a strong point M then the locus seems to be a circular cK(#X1,R) with singular focus F.

When M = X2, you get K086.

When M = X6, you get K040.

[APH]

For the case of the cyclologic center (ABC, PaPbPc) M can be any point strong or not.
That is:
Let ABC be a triangle, IaIbIc the excentral triangle, M a fixed point and P a point on the line OM.

Denote:

Pa, Pb,Pc = same to P points of IaBC,IbCA,IcAB, resp.

The triangles ABC, PaPbPc are circumcyclologic.

As P moves on the line OM:
The cyclologic center (ABC, PaPbPc) is a fixed point Q on the circumcircle.
The locus of the cyclologic center (PaPbPc, ABC) is a cubic.

Let's see the Q's. The class of the cubics is a subject of Bernard Gibert.
1. P on the Euler line
Q = X(100) = Reflection point of IO = X(1)X(3) line = Reflection point of Euler line of INTOUCH triangle (pedal triangle of I).

2. P on the Brocard axis
Q = X(101) = Reflection point of X(1)X(7) line = Reflection point of Brocard axis of INTOUCH triangle (pedal triangle of I).

Generalization

Let M be a fixed Point and P be a point on the line OM = L
Denote: P' = the same to P point of the INTOUCH triangle.
L' = the same to L line of the INTOUCH triangle.

Then Q is the reflection point of the L' line = IP' line of ABC

Note:
Reflection point of a line L:
The reflections La, Lb, Lc of L in the siedelines BC, CA, AB, resp. bound a triangle A*B*C*.
ABC, A*B*C* are perspective.The perspector, lying on the circumcircle, is called "Reflection pont of the line L"
It is the incenter (or an excenter) of the triangle A*B*C*.

Case of OM with M = I= X(1)

1. P = X(1) = O of INTOUCH triangle.
Pa, Pb, Pc = X(1) of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q

2. P = X(354) = X(2) of the INTOUCH triangle
Pa, Pb, Pc = X(2) of the intouch triangles of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q1

3. P = X(942) = X(5) of the INTOUCH triangle
Pa, Pb, Pc = X(5) of the intouch triangles of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q2

Q, Q1, Q2 coincide.

Q is the reflection point of the IO line of the INTOUCH triangle.
It is the line passing thrpugh the incenters of ABC and the intouch triangle.

A CYCLOLOGIC THEOREM RELATED TO ORTHIC TRIANGLE

[APH]

Orthic Version

Let ABC be a triangle, HaHbHc the orthic triangle and P a point.

Denote:

Pa, Pb, Pc = same to P points of AHbHc, BHcHa, CHaHb, resp.

HaHbHc, PaPbPc are circumcyclologic

Cyclologic center (HaHbHc, PaPbPc = ? (on the circumcircle of HaHbHc = NPC)
Cyclologic center (PaPbPc, HaHbHc) = ? (on the circumcircle of PaPbPc)

[Ercole Suppa]

cyclologic center (HaHbHc, PaPbPc) = Poncelet point(isogonal conjugate(P))

cyclologic center (PaPbPc, HaHbHc) = orthoassociate(isogonal conjugate(circumcircleInverse(P)))

Euclid 10026

LOCUS PROBLEM

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán

ETC LISTING OF Q

X(72803)

FOUR POINTS ON THE EULER LINE

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán
FOUR POINTS ON THE EULER LINE

ETC LISTINGS

X(72698)

X(72699)

X(72700)

X(72701)

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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