LOCUS PROBLEM

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán

ETC LISTING OF Q

X(72803)

ETC

X(72882) = X(1)X(527)∩X(36)X(14074)

Barycentrics    (a^2+4*a*b+b^2-2*a*c-2*b*c+c^2)*(a^2-2*a*b+b^2+4*a*c-2*b*c+c^2)*(4*a^3-7*a^2*b+2*a*b^2+b^3-7*a^2*c+6*a*b*c-b^2*c+2*a*c^2-b*c^2+c^3) : :

Antreas Hatzipolakis and Ercole Suppa, euclid 10075.

X(72882) lies on the cubic K086 and these lines: {1, 527}, {36, 14074}, {28292, 46919}

X(72882) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {46919, 1, 4419}
X(72882) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1),X(53181)}, {A,B,C,X(80),X(527)}, {A,B,C,X(28292),X(64262)}, {A,B,C,X(47357),X(53183)}}
X(72882) = X(i)-isoconjugate of X(j) for these {i,j}: {14077, 53887},{37541, 72681}
X(72882) = barycentric product X(i)*X(j) for these (i,j): {34919, 50573}
X(72882) = trilinear quotient X(i)/X(j) for these (i,j): {8545, 50573}, {14074, 53887}, {34919, 72681}

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X(72883) = X(1)X(527)∩X(1319)X(3321)

Barycentrics    10*a^7-5*a^6*b-66*a^5*b^2+115*a^4*b^3-50*a^3*b^4-15*a^2*b^5+10*a*b^6+b^7-5*a^6*c+84*a^5*b*c-99*a^4*b^2*c-56*a^3*b^3*c+93*a^2*b^4*c-12*a*b^5*c-5*b^6*c-66*a^5*c^2-99*a^4*b*c^2+228*a^3*b^2*c^2-78*a^2*b^3*c^2-42*a*b^4*c^2+9*b^5*c^2+115*a^4*c^3-56*a^3*b*c^3-78*a^2*b^2*c^3+88*a*b^3*c^3-5*b^4*c^3-50*a^3*c^4+93*a^2*b*c^4-42*a*b^2*c^4-5*b^3*c^4-15*a^2*c^5-12*a*b*c^5+9*b^2*c^5+10*a*c^6-5*b*c^6+c^7 : :

Antreas Hatzipolakis and Ercole Suppa, euclid 10075.

X(72883) lies on these lines: {1, 527}, {1319, 3321}, {1638, 28292}

X(72883) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {1638, 1, 4419}

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X(72884) = X(1)X(5180)∩X(36)X(5606)

Barycentrics    (a^3+3*a^2*b+3*a*b^2+b^3-a^2*c-a*b*c-b^2*c-a*c^2-b*c^2+c^3)*(a^3-a^2*b-a*b^2+b^3+3*a^2*c-a*b*c-b^2*c+3*a*c^2-b*c^2+c^3)*(2*a^4-a^3*b-3*a^2*b^2+a*b^3+b^4-a^3*c-4*a^2*b*c+4*a*b^2*c-3*a^2*c^2+4*a*b*c^2-2*b^2*c^2+a*c^3+c^4) : :

Antreas Hatzipolakis and Ercole Suppa, euclid 10075.

X(72884) lies on these lines: {1, 5180}, {36, 5606}, {31947, 44824}

X(72884) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {31947, 1, 5180}, {1, 31947, 44824}
X(72884) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(79),X(65999)}, {A,B,C,X(1255),X(5483)}, {A,B,C,X(12913),X(61105)}}
X(72884) = pole of the line {7332, 26842} with respect to dual conic of Yff parabola
X(72884) = inverse of X(18244) in incircle
X(72884) = X(i)-isoconjugate of X(j) for these {i,j}: {6584, 8702},{6595, 14882}
X(72884) = trilinear quotient X(i)/X(j) for these (i,j): {5606, 6584}, {6595, 10266}

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X(72885) = X(1)X(6692)∩X(36)X(30206)

Barycentrics    (a^3-a^2*b-a*b^2+b^3-3*a^2*c+8*a*b*c-b^2*c-3*a*c^2-b*c^2+c^3)*(a^3-3*a^2*b-3*a*b^2+b^3-a^2*c+8*a*b*c-b^2*c-a*c^2-b*c^2+c^3)*(4*a^4-5*a^3*b-3*a^2*b^2+5*a*b^3-b^4-5*a^3*c+16*a^2*b*c-7*a*b^2*c-3*a^2*c^2-7*a*b*c^2+2*b^2*c^2+5*a*c^3-c^4) : :

Antreas Hatzipolakis and Ercole Suppa, euclid 10075.

X(72885) lies on the cubic K086 and these lines: {1, 6692}, {36, 30236}, {1785, 61484}, {1795, 61483}

X(72885) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1785),X(61483)}, {A,B,C,X(1795),X(61484)}, {A,B,C,X(10309),X(66217)}, {A,B,C,X(30719),X(63621)}}
X(72885) = X(i)-isoconjugate of X(j) for these {i,j}: {30198, 53897},{41426, 56097}
X(72885) = trilinear quotient X(i)/X(j) for these (i,j): {30236, 53897}, {56089, 56097}

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