HYACINTHOS 28987

[Kadir Altintas - Ercole Suppa]: 

 

Let ABC be a triangle, P a point and  DEF the pedal triangle of P

Let A' be the second intersection of PD with the circle (DEF)

The circle with diameter A'P intersects the circle (DEF) again at  A''

Define B'',C'' cyclically

 

Prove: the triangle A''B''C'' is perspective with ABC

 

Romantics of Geometry, problem 2968 

 

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[Ercole Suppa]

 

Let Q be the perspector between ABC and A''B''C''. We have:

 

 

*** Paιrs (P = X(i),Q = X(j)) for these {i,j}: {1,8}, {4,6526}, {74,18808}, {99,877}, {109,23987}, {110,4240}, {112,23977}

 

*** Some points:

 

 

Q(X(2)) = (name pending)

 

= (a^2+b^2-5 c^2) (a^2-5 b^2+c^2) (a^4-16 a^2 b^2+7 b^4+14 a^2 c^2-16 b^2 c^2+c^4) (a^4+14 a^2 b^2+b^4-16 a^2 c^2-16 b^2 c^2+7 c^4) : : (barys)

 

= (108 R^2 S^2+18 S^2 SB-21 S^2 SW-4 SW^3) (108 R^2 S^2+18 S^2 SC-21 S^2 SW-4 SW^3) : : (barys)

 

= lies on Kiepert hyperbola and this line: {4,17952}

= isogonal conjugate of Q*(X(2))

 

= barycentric quotient of X(i) and X(j) for these {i,j}: {5485, 9741}

 

= (6-9-13) search numbers [5.48823320479735349, 1.12308230287118749, 0.330115254628624617]

 

 

Q*(X(2)) = (name pending)  

 

= a^2 (5 a^2-b^2-c^2) (7 a^4-16 a^2 b^2+b^4-16 a^2 c^2+14 b^2 c^2+c^4) : : (barys)

 

= 18 S^4 + (108 R^2 SB+108 R^2 SC-18 SB SC-21 SB SW-21 SC SW)S^2 -4 SB SW^3-4 SC SW^3 : : (barys) 

 

= lies on this line: {3,6}

 = isogonal conjugate of Q(X(2))

= barycentric product of X(i) and X(j) for these {i,j}: {1384, 9741}

 

= (6-9-13) search numbers [0.177854679900480515, 0.869133061186146071, 2.95687020267067159]

 

 

Q(X(3)) = X(2)X(254) ∩ X(3)X(2165)

 

= (a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-2 b^2 c^2+c^4) (a^4+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2+2 a^2 b^2 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6) : : (barys)

 

= (6 R^2-SW)S^4 + (-8 R^6+8 R^4 SB+8 R^4 SC+2 R^2 SB SC+8 R^4 SW-6 R^2 SB SW-6 R^2 SC SW-SB SC SW-2 R^2 SW^2+SB SW^2+SC SW^2)S^2 + 8 R^6 SB SC-8 R^4 SB SC SW+2 R^2 SB SC SW^2  : : (barys)

 

= lies on these lines: {2,254}, {3,2165}, {24,16172}, {68,394}, {96,97}, {925,3147}, {3546,5392}, {3926,20563}, {6642,14593}

= barycentric product of X(i) and X(j) for these {i,j}: {68,6504}, {5392,15316}

 

= barycentric quotient of X(i) and X(j) for these {i,j}: {68,6515}, {115,135}, {254,11547}, {921,1748}, {1820,920}, {2165,3542}, {2351,1609}, {6504,317},{8800,467}, {15316,1993}, {16391,6503}

 

= trilinear product of X(i) and X(j) for these {i,j}: {68, 921}, {68, 921}, {91, 15316}, {1820, 6504}, {1820, 6504}

 

= trilinear quotient of X(i) and X(j) for these {i,j}: {68,920}, {91,3542}

 

= (6-9-13) search numbers [6.36369648663114288, 0.882338833648135950, 0.0927238340136036553]

 

 

 

Best regards

Ercole Suppa