[Aris Pavlakis] (*)
Let M1, M2, M3 be three random points on the sides BC, CA, AB, resp. of triangle ABC.
Let (K) be the circle (M1M2M3) and K1, K2, K3 the orthogonal projections of K on BC, CA, AB, resp.
Let A1 = MM1 /\ KK1, B1 = MM2 /\ KK2, C1 = MM3 /\ KK3
Let N1, N2, N3 be the other (than M1, M2, M3) intersections of (K) and BC, CA, AB, resp.
1. ABC, A1B1C1 are similar.
2. A1N1, B1Ν2. C1N3 are concurrent.
(*) Romantics of Geometry 3364
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If M1M2M3 is pedal triangle of a point P, then P = M and A1, B1, C1 lie on the line at infinity,
Rewritting the concurrence part.
Let ABC be a triangle and M1, M2, M3 three points on BC, CA, AB, resp.
Denote:
M = the Miquel point of M1M2M3
K = the circumcenter of M1M2M3
K1K2K3 = the pedal triangle of K
Then the reflections of MM1, MM2, MM3 in KK1, KK2, KK3, resp. are concurrent.
Which is the point of concurrence if M1M2M3 is the cevian triangle of a point P?
Is it the isogonal conjugate of M?
[Ercole Suppa]:
Hi Antreas
If P(x:y:z) (barys) then the reflections of MM1, MM2, MM3 in KK1, KK2, KK3, resp. concur at point
Q = Q(P) = (y+z) (-2 b^2 c^2 x^2 y (x+y) z (x+z) (y+z)^2+c^4 x^2 y^2 (x+z)^2 (y+z)^2-(x+y)^2 z^2 (a^4 y^2 (x+z)^2-b^4 x^2 (y+z)^2)) : :
which is the isogonal conjugate of M (Miquel associate of P).
*** Pairs (P = X(i), Q(P) = X(j)): {1,502},{2,4},{4,3},{7,1},{8,84},{20,3346},{30,2133},{69,64},{99,6328},{189,40},{253,20},{254,15242},{264,6798},{329,3345},{512,2143},{1029,501},{1031,14885},{1032,1498},{1034,1490},{1138,1117},{5932,3347},{8047,11607},{13485,14366},{13582,14354},{14361,3348},{14362,2131},{14365,3183}
*** Some points:
Q(X(3)) = X(5)X(3462) ∩ X(30)X(3481)
= (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^12+3 a^10 b^2-a^8 b^4-6 a^6 b^6+9 a^4 b^8-5 a^2 b^10+b^12+3 a^10 c^2-5 a^8 b^2 c^2+2 a^6 b^4 c^2-6 a^4 b^6 c^2+11 a^2 b^8 c^2-5 b^10 c^2-3 a^8 c^4+2 a^6 b^2 c^4-2 a^4 b^4 c^4-6 a^2 b^6 c^4+9 b^8 c^4+2 a^6 c^6+2 a^4 b^2 c^6+2 a^2 b^4 c^6-6 b^6 c^6-3 a^4 c^8-5 a^2 b^2 c^8-b^4 c^8+3 a^2 c^10+3 b^2 c^10-c^12) (a^12-3 a^10 b^2+3 a^8 b^4-2 a^6 b^6+3 a^4 b^8-3 a^2 b^10+b^12-3 a^10 c^2+5 a^8 b^2 c^2-2 a^6 b^4 c^2-2 a^4 b^6 c^2+5 a^2 b^8 c^2-3 b^10 c^2+a^8 c^4-2 a^6 b^2 c^4+2 a^4 b^4 c^4-2 a^2 b^6 c^4+b^8 c^4+6 a^6 c^6+6 a^4 b^2 c^6+6 a^2 b^4 c^6+6 b^6 c^6-9 a^4 c^8-11 a^2 b^2 c^8-9 b^4 c^8+5 a^2 c^10+5 b^2 c^10-c^12) : : (barys)
= (SA SB+SA SC+2 SB SC) (SA^3 SB^3+SA^3 SB^2 SC+SA^2 SB^3 SC-SA^3 SB SC^2-5 SA^2 SB^2 SC^2-SA SB^3 SC^2-SA^3 SC^3-5 SA^2 SB SC^3-5 SA SB^2 SC^3-SB^3 SC^3) (SA^3 SB^3+SA^3 SB^2 SC+5 SA^2 SB^3 SC-SA^3 SB SC^2+5 SA^2 SB^2 SC^2+5 SA SB^3 SC^2-SA^3 SC^3-SA^2 SB SC^3+SA SB^2 SC^3+SB^3 SC^3) : : (barys)
= lies on these lines: {5,3462}, {30,3481}
= barycentric product of X(i) and X(j) for these {i,j}: {264,21354}, {324,3463}
= barycentric quotient of X(i) and X(j) for these {i,j}: {53,3462}, {3463,97}, {21354,3}
= trilinear product X(92)*X(21354)
= trilinear quotient X(1087)/X(15780)
= (6-9-13) search numbers: [0.1165290910764150415, 0.0814670196188231348, 3.5304815801360792214]
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Q(X(6)) = COMPLEMENT OF X(13511)
= (b^2+c^2) (-a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2-c^4) (a^4+a^2 b^2+b^4+a^2 c^2+b^2 c^2-c^4) : : (barys)
= (2 SA+SB+SC) (SA^2+5 SA SB+3 SB^2+SA SC+5 SB SC+SC^2) (SA^2+SA SB+SB^2+5 SA SC+5 SB SC+3 SC^2) : : (barys)
= lies on these lines: {2,13500}, {5,30218}, {141,384}, {3051,14886}, {4074,7794}, {10007,12143}, {28667,28677}
= complement of X(13511)
= isogonal conjugate of X(14885)
= isotomic conjugate of isogonal conjugate of X(21355)
= barycentric product of X(i) and X(j) for these {i,j}: {38,18834}, {76,21355}, {141,1031}, {8024,14370}
= barycentric quotient of X(i) and X(j) for these {i,j}: {6,14885}, {38,16556}, {39,10329}, {141,2896}, {1031,83}, {1930,20934}, {3665,17083}, {3917,22138}, {3954,21880}, {14370,251}, {15523,21083}, {16892,21194}, {18834,3112}, {21355,6}
= trilinear product of X(i) and X(j) for these {i,j}: {38,1031}, {38,1031}, {39,18834}, {39,18834}, {75,21355}, {75,21355}, {1930,14370}, {1930,14370}
= trilinear quotient of X(i) and X(j) for these {i,j}: {1,14885}, {38,10329}, {141,16556}, {1930,2896}, {8024,20934}, {15523,21880}
= (6-9-13) search numbers: [0.5144793024262912038, 0.3456511786717395506, 3.1639155263225788586]
Best regards,
Ercole Suppa
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