Dear all,
Let
* a triangle ABC and a point P on the plane
* A_0B_0C_0: cevian triangle of P w.r.t ABC
* Xba,Yba: the incenter and the P-excenter of of triangles PBA_0
Xca,Yca the incenter and the P-excenter of of triangles PcA_0
define ,Xcb,Ycb,Xab,Yab,Xac,Yac,Xbc,Ybc similarly
* three lines XbaXca , XcbXab, XacXbc bounds a triangle A_xB_xC_x
* three lines YbaYca , YcbYab, YacYbc bounds a triangle A_yB_yC_y
Then :
1. triangles ABC and A_xB_xC_x are perspective with perspector X ?
2. triangles ABC and A_yB_yC_y are perspective with perspector Y ?
2. triangles A_xB_xC_x and A_yB_yC_y are perspective with perspector P' ?
4. X,Y,P' are collinear ?
Best regards,
Vu Thanh Tung & Vu Quoc My
[César Lozada]:
Let’s rewrite because there is a new condition (underlined here) in Anopolis message #9113:
Let ABC be a triangle, P a point interior to ABC and A0B0C0 the cevian triangle of P.
Denote:
Iba = the incenter of PBA0, and cyclically Icb and Iac
Ica = the incenter of PCA0, and cyclically Iab and Ibc
Eba = the P-excenter of PBA0, and cyclically Ecb and Eac
Eca = the P-excenter of PCA0, and cyclically Eab and Ebc
AiBiCi = the triangle bounded by the lines { Iba , Ica }, { Icb , Iab }, { Iac , Ibc }
AeBeCe = the triangle bounded by the line { Eba , Eca }, { Ecb , Eab }, { Eac , Ebc }
Then:
1) AiBiCi and ABC are perspective (with perspector Qi).
2) AeBeCe and ABC are perspective (with perspector Qe).
3) AiBiCi and AeBeCe are perspective (with perspector Qie).
4) Qi, Qe and Qie are collinear (on the polar trilinear of Q*)
-----------------------------
Let P=u:v:w (exact trilinears) and pa, pb, pc the sidelenghts of the pedal triangle of P w/r to ABC, ie,
pa = sqrt(v^2 + w^2 + 2*v*w*cos(A)) and cyclically pb, pc. Then:
1) Qi = 1/(v*w + u* pa) : : (trilinears)
2) Qe = 1/(v*w - u* pa) : : (trilinears)
3) Qie = (u^3*pa^2+v^3*pb^2+w^3*pc^2-pa*u*((u+v)*v*pb+w*(u+w)*pc)-(v+w)*v*w*pb*pc)*u^2*v*w+(v^2*pb-w^2*pc)*(v*pb-w*pc)*v*w*pb*pc+(-(u*pa+v*pb+w*pc)*v*w*pb*pc+((2*u+w)*v^3*pb+(2*u+v)*w^3*pc)*u)*pa*pb*pc : : (trilinears)
4) Q* = 1/(u*(v^2*pb-w^2*pC)*(v^2*w^2-u^2*pa^2)) : : (trilinears)
Some related centers:
Notes:
If P=X(3) then pa = a/2
If P=X(4) the pa = |a*cos(A)|
---------------------------------------
For P=X(3)
P0 = SS(S → -S) of X(8954)
= a*((-S+SB)*b+(-S+SC)*c+a*b*c) :: (barys)
= lies on these lines: {3, 6}, {24, 6414}, {155, 10960}, {485, 6810}, {486, 1586}, {590, 8955}, {1075, 3069}, {1600, 26922}, {5562, 8963}, {6413, 7592}, {6565, 8887}, {6642, 10962}, {8855, 15199}, {10880, 26920}
= SS(S → -S) of X(8954)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (216, 389, 8954), (3371, 3372, 577), (3385, 3386, 578)
= [ 1.4167409784956670, 1.8118856009409550, 1.7324016911811030 ]
P1 = X(1)X(372) ∩ X(4)X(48)
= a*( 2*(-a^2+b^2+c^2)*S*a^2+(a^2+b^2-c^2)*(a^2-b^2+c^2)*b*c) :: (barys)
= lies on these lines: {1, 372}, {4, 48}, {41, 1588}, {101, 31562}, {572, 31561}, {604, 1587}, {610, 6212}, {637, 20769}, {1745, 2067}, {2066, 3362}, {2174, 3071}, {2286, 3093}, {3070, 7113}, {3092, 7124}, {5709, 19215}, {7330, 19216}
= isogonal conjugate of P2
≈ [ 2.8354085702879280, 2.8249064679139530, 2.7716950362187460 ] (9-10-13 search)
≈ [ -0.1417181308130000, 0.4658221516300000, 3.3835805911540000 ] (6-9-13 search)
P2 = Qi = ISOGONAL CONJUGATE OF P1
= a*(-(a^2-b^2+c^2)*(a^2-b^2-c^2)*a*b+2*(a^2+b^2-c^2)*S*c^2)*(-(a^2-b^2-c^2)*(a^2+b^2-c^2)*a*c+2*(a^2-b^2+c^2)*S*b^2) : : (barys)
= a*(S^3+(a*b-S)*SA*SB)*(S^3+SA*(-S+c*a)*SC) : : (barys)
= sin(A)*(2*cos(A)*cos(C)+sin(2*B))*(2*cos(A)*cos(B)+sin(2*C)) : : (trilinears)
= lies on the lines: {1, P0}, {1745, 6212}
= isogonal conjugate of P1
≈ [2.7770827567684531077, 2.7874070658186284113, 2.8409201394978639363] (9-10-13 search)
≈ [ 17.4167765035310000, -5.2987454594680000, -0.7294855093210000 ] (6-9-13 search)
P3 = X(1)X(371) ∩ X(4)X(48)
= a*(-2*(-a^2+b^2+c^2)*S*a^2+(a^2+b^2-c^2)*(a^2-b^2+c^2)*b*c) :: (barys)
= lies on these lines: {1, 371}, {4, 48}, {41, 1587}, {101, 31561}, {572, 31562}, {604, 1588}, {610, 6213}, {638, 20769}, {1745, 6502}, {2174, 3070}, {2286, 3092}, {3071, 7113}, {3093, 7124}, {3362, 5414}, {5709, 19216}, {7330, 19215}
= isogonal conjugate of P4
≈ [ 2.8354085702879280, 2.8249064679139530, 2.7716950362187460 ] (9-10-13 search)
≈ [ -16.8316420438520000, -20.6328686943790000, 25.6934083674860000 ] (6-9-13 search)
P4 = Qe = ISOGONAL CONJUGATE OF P3
= a*(-(a^2-b^2+c^2)*(a^2-b^2-c^2)*a*b-2*(a^2+b^2-c^2)*S*c^2)*(-(a^2-b^2-c^2)*(a^2+b^2-c^2)*a*c-2*(a^2-b^2+c^2)*S*b^2) : : (barys)
= a*(S^3+(a*b+S)*SA*SB)*(S^3+SA*(a*c+S)*SC) : : (barys)
= sin(A)*(2*cos(A)*cos(C)-sin(2*B))*(2*cos(A)*cos(B)-sin(2*C)) : : (trilinears)
= lies on the lines: {1, 8954}, {1745, 6213}
= isogonal conjugate of P3
≈ [ 20.7142720294590000, 21.3186470840790000, -23.8319339864110000 ] (9-10-13 search)
≈ [ 9.8076801908110000, 8.0007954636609990, -6.4249693886940000 ] (6-9-13 search)
Qie : Long and not interesting
Q* : Not interesting
--------------------------------------
For P=X(4)
Qi =X(485)
Qe =X(486)
The line { X(485), X(486) } = { X(5), X(6) }
P5 = Qie = X(5)X(6) ∩ X(485)X(486)
= a^9-(b+c)*a^8-(b^2+c^2)*a^7+(b+c)*(3*b^2-2*b*c+3*c^2)*a^6-(b^2-c^2)^2*a^5-(b+c)*(5*b^4+5*c^4-6*(b^2-b*c+c^2)*b*c)*a^4+(b^4-c^4)*(b^2-c^2)*a^3+(b^2-c^2)*(b-c)*(5*b^4+5*c^4+2*(2*b^2+b*c+2*c^2)*b*c)*a^2-2*(b^3-c^3)*(b^2-c^2)^3 : : (barys)
= lies on these lines: {3, 8287}, {5, 6}, {368, 18453}, {1736, 1893}, {2478, 25000}, {5221, 5729}, {5706, 14873}, {5740, 6835}
≈ [7.2785202642988065, 0.0588468354553874, 1.8234328114870355] (9-10-13 search)
≈ [1.0905025320838760, 1.8902343413290780, 1.8287318461793830] (6-9-13 search)
Q*= X(925)
César Lozada
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