LOCUS PROBLEM

1.
Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán

LOCUS PROBLEM 1

2.
Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán

LOCUS PROBLEM 2

ETC POINTS

X(73252) = X(2)X(490)∩X(3)X(6118)

Barycentrics    -a^4+4*a^2*b^2-3*b^4+4*a^2*c^2+6*b^2*c^2-3*c^4+2*a^2*sqrt(-a^4+2*a^2*b^2-b^4+2*a^2*c^2+2*b^2*c^2-c^4)+b^2*sqrt(-a^4+2*a^2*b^2-b^4+2*a^2*c^2+2*b^2*c^2-c^4)+c^2*sqrt(-a^4+2*a^2*b^2-b^4+2*a^2*c^2+2*b^2*c^2-c^4) : :
X(73252) = 3*X[2]+X[1131], 3*X[2]-X[33364], 9*X[2]-X[51953], X[1131]+X[33364], 3*X[1131]+X[51953], 3*X[33364]-X[51953], 4*X[6118]-X[15293], X[42838]-2*X[45871], X[6568]-5*X[14061]

Fotis Dellaportas and Francisco Javier García Capitán, euclid 10523.

X(73252) lies on this line: {2, 490}, {3, 6118}, {4, 42838}, {5, 13748}, {69, 43879}, {115, 13882}, {371, 5461}, {381, 64691}, {485, 32491}, {487, 590}, {488, 53513}, {524, 13886}, {591, 7583}, {637, 13846}, {639, 1991}, {641, 13665}, {1151, 32489}, {1656, 6119}, {2023, 13773}, {3068, 23311}, {3071, 39143}, {3090, 3589}, {3103, 3934}, {3594, 32807}, {3618, 42583}, {5055, 45378}, {5067, 45872}, {5418, 13835}, {6036, 13749}, {6230, 6289}, {6304, 6307}, {6305, 6306}, {6329, 13939}, {6564, 11315}, {6568, 14061}, {6669, 18587}, {6670, 18586}, {6704, 11314}, {7375, 69139}, {7389, 8253}, {8294, 44531}, {11291, 32789}, {11292, 42273}, {12322, 31454}, {13903, 32419}, {13972, 31404}, {22596, 37343}, {22717, 49351}, {23251, 39387}, {31534, 58463}, {32988, 49229}, {37637, 53487}, {37691, 57266}, {38317, 44509}, {41963, 58804}, {43880, 51171}, {45484, 49221}

X(73252) = midpoint of X(1131) and X(33364)
X(73252) = reflection of X(42838) in X(45871)
X(73252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1131, 33364}, {485, 32491, 45472}, {639, 8976, 1991}, {10576, 11313, 45473}

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X(73253) = X(2)X(489)∩X(3)X(6119)

Barycentrics    a^4+4*a^2*b^2-3*b^4+4*a^2*c^2+6*b^2*c^2-3*c^4-2*a^2*sqrt(-a^4+2*a^2*b^2-b^4+2*a^2*c^2+2*b^2*c^2-c^4)-b^2*sqrt(-a^4+2*a^2*b^2-b^4+2*a^2*c^2+2*b^2*c^2-c^4)-c^2*sqrt(-a^4+2*a^2*b^2-b^4+2*a^2*c^2+2*b^2*c^2-c^4) : :
X(73253) = 3*X[2]+X[1132], 3*X[2]-X[33365], 9*X[2]-X[51952], X[1132]+X[33365], 3*X[1132]+X[51952], 3*X[33365]-X[51952], 4*X[6119]-X[15294], X[42840]-2*X[45872], X[6569]-5*X[14061]

Fotis Dellaportas and Francisco Javier García Capitán, euclid 10523.

X(73253) lies on this line: {2, 489}, {3, 6119}, {4, 42840}, {5, 13749}, {69, 43880}, {115, 13934}, {372, 5461}, {486, 32490}, {487, 53516}, {488, 615}, {524, 13939}, {591, 640}, {638, 13847}, {642, 13785}, {1152, 32488}, {1592, 55477}, {1656, 6118}, {1991, 7584}, {2023, 13653}, {3069, 23312}, {3070, 39143}, {3090, 3589}, {3102, 3934}, {3591, 32814}, {3618, 42582}, {5055, 45377}, {5067, 45871}, {5070, 64691}, {5420, 13712}, {6036, 13748}, {6231, 6290}, {6300, 6303}, {6301, 6302}, {6329, 13886}, {6565, 11316}, {6569, 14061}, {6669, 18586}, {6670, 18587}, {6704, 11313}, {7376, 69139}, {7388, 8252}, {8293, 44531}, {11291, 42270}, {11292, 32790}, {13910, 31404}, {13961, 32421}, {22625, 37342}, {22719, 49352}, {23261, 39388}, {31454, 32813}, {31535, 58463}, {32807, 59635}, {32988, 49228}, {37637, 53488}, {37691, 57267}, {38317, 44510}, {41964, 58803}, {43879, 51171}, {45485, 49220}

X(73253) = midpoint of X(1132) in X(33365)
X(73253) = reflection of X(42840) in X(45872)
X(73253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1132, 33365}, {486, 32490, 45473}, {640, 13951, 591}, {10577, 11314, 45472}

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