Let ABC be a triangle.
P is any point.
The parallel line from P to BC meets the circles (PAB), (PAC) at Bc, Cb, respectively.
Define similarly the points Ca,Ac, Ba, Ab.
Then
1) The perpendicular bisectors of the segments BcCB, AcCa, AbBa are concurrent.
2) The lines BaCa, AbCb and BcAc bound a triangle which is perspective to ABC.
3) The lines AbAc, BcBa, CaCb bound a triangle which is perspective to ABC.
Which are the concurrence points?
[César Lozada]:
Assume P=x:y:z (barys)
1) Perspector Q(P)
Q(P) = (x+y+z)*a*b*c*x*y*z*cos(A)+x^2*(c^3*cos(B)*y^2+z^2*cos(C)*b^3)-a^3*y^2*z^2 : :
If P lies on the circumcircle of ABC then Q(P)=antipode-of-P
Other ETC pairs (P,Q(P)): (1,3), (2,376), (3,6759), (4,4), (6,182), (7,2550), (8,12667), (13,5617), (14,5613), (20,3183), (57,999), (59,1618), (64,13346), (80,10742), (84,10306), (265,19506)
Some non-ETC:
Q(X(5)) = X(3)X(128) ∩ X(4)X(252)
= 5*S^4+(R^2*(4*R^2-5*SA)+2*SA^2-7*SB*SC-SW^2)*S^2+(R^2*(20*R^2-19*SW)+5*SW^2)*SB*SC : : (barys)
= 3*X(5)-2*X(20414)
= on lines: {3, 128}, {4, 252}, {5, 6150}, {30, 14143}, {550, 6247}, {930, 2888}, {933, 3462}, {1510, 11591}, {3153, 14097}, {3574, 12060}, {15619, 15704}
= midpoint of X(15619) and X(15704)
= {X(3), X(1601)}-harmonic conjugate of X(23320)
= [ 12..9091550165322000, 15.5810471118121600, -13.1042858339004300 ]
Q(X(15)) = X(3)X(24303) ∩ X(61)X(16461)
= (SB+SC)*(3*(6*R^2+2*SA-3*SW)*S^2-sqrt(3)*(S^2-36*R^2*SA+5*SA^2+4*SB*SC)*S+3*(6*R^2-3*SA+2*SW)*SA*SW) : : (barys)
= 2*X(13350)-3*X(14170)
= on lines: {3, 24303}, {61, 16461}, {616, 10409}, {1495, 13350}, {5663, 13859}
= [ 241.0949715056529000, 275.4802155261222000, -298.3510100387862000 ]
Q(X(16)) = X(3)X(24304) ∩ X(62)X(16462)
= (SB+SC)*(3*(6*R^2+2*SA-3*SW)*S^2+qrt(3)*(S^2-36*R^2*SA+5*SA^2+4*SB*SC)*S+3*(6*R^2-3*SA+2*SW)*SA*SW) : : (barys)
= on lines: {3, 24304}, {62, 16462}, {617, 10410}, {1495, 13349}, {5663, 13858}
= [ -4.5434787324335270, -12.5756903632501900, 14.4439018406653700 ]
Q(X(31)) = X(3)X(695) ∩ X(192)X(815)
= a^2*(b*c*a^6-(b^3+c^3)*(b+c)*a^4+(b^3+c^3)*b*c*a^3+(b^4-c^4)*(b^2-c^2)*a^2-(b^3+c^3)*(b^2+c^2)*b*c*a+2*b^4*c^4) : : (barys)
= on lines: {3, 695}, {192, 815}, {6310, 19548}
= [ 1.0498079147185610, 2.1906983948018270, 1.6395004017899320 ]
2) Perspector Q’(P):
Q`(P) = F(b,c,a,y,z,x) * F(c,a,b,z,x,y) : :
Where
F(a,b,c,x,y,z) = (z^2*b^2+y^2*c^2)*b^2*c^2*x^4+a^2*x^3*(c^4*y^3+b^4*z^3)+a^2*x^2*y*z*(y^2*c^4+z^2*b^4)+x^3*(-a^2+b^2+c^2)*y*z*((z*b^2+c^2*y)*a^2+b^2*c^2*x)+(-a^4+b^4+c^4)*a^2*x^2*y^2*z^2-a^4*y^2*z^2*(a^2*y*z+x*(y*(a^2-b^2)+z*(a^2-c^2)))
- If P lies on the circumcircle of ABC then Q’(P)=isogonal-of-P
- If P lies in the infinity then Q’(P) also lies in the infinity.
ETC-pairs (P,Q’(P)) for P in the infinity: (511,804), (512,804), (513,900), (514,926), (516,926), (517,900), (518,6084),
(523,526), (524,6088), (690,20403), (758,6089), (1499,6088), (1510,25149), (3309,6084), (3667,6085)
- Other ETC-pairs (P,Q’(P)): (1,4), (3,64), (4,3), (6,13377)
Q’(X(2)) = ISOGONAL CONJUGATE OF X(6031)
= a^2*(2*(2*b^2+c^2)*a^4+(b^2+2*c^2)*a^2*c^2-4*(b^4-c^4)*b^2)*(2*(b^2+2*c^2)*a^4+(2*b^2+c^2)*a^2*b^2+4*(b^4-c^4)*c^2) : : (barys)
= on lines: {6, 6324}, {574, 12367}, {599, 8705}
= reflection of X(6) in X(6324)
= anticomplement of the complementary conjugate of X(6032)
= antigonal conjugate of the isogonal conjugate of X(5971)
= isogonal conjugate of X(6031)
= [ -9.1419909761642140, -15.8907629561694100, 18.8613423636389900 ]
3) Perspector Q”(P):
Q”(P) = y*z*(b^2*z+c^2*y)*((x+z)*a^2*z-(b^2*z-c^2*x-c^2*z)*x)*((x+y)*a^2*y+(b^2*x+b^2*y-c^2*y)*x) : :
- If P lies on the circumcircle of ABC then Q”(P)=isogonal-of-P
- ETC-pairs (P,Q”(P)) for P in the infinity: (30,14264), (511,14265), (512,76), (513,8), (514,3730), (517,14266), (518,14267), (520,14249), (521,14257), (522,10571), (523,3), (524,14263), (525,8743), (526,14254), (690,14246), (804,14251), (826,14247), (900,14260), (924,847), (1499,14262), (1510,25043), (3309,14268), (3566,14248), (3667,14261), (3800,14259), (3900,14256), (7927,14250), (8678,14258), (9023,14255), (23878,14252)Other ETC-pairs (P,Q’(P)): (1,4), (3,64), (4,3), (6,13377)
- Other ETC-pairs (P,Q”(P)): (1,65), (3,4), (4,5), (6,598), (15,8014), (16,8015), (36,14584), (64,18848), (186,14254), (187,598)
Q”(X(2)) = ISOGONAL CONJUGATE OF X(10130)
= a^2*(b^2+c^2)*(2*a^2+2*b^2-c^2)*(2*a^2+2*c^2-b^2) : : (barys)
= on lines: {6, 23}, {39, 9019}, {76, 524}, {141, 23297}, {230, 25488}, {523, 18907}, {597, 13410}, {755, 11636}, {882, 9009}, {2353, 30435}, {2393, 27375}, {2854, 5052}, {3291, 16776}, {3629, 6664}, {6698, 15820}, {7737, 11594}, {8584, 20380}, {9465, 9971}
= isogonal conjugate of X(10130)
= [ 0.4090224123381342, 0.7095357786310635, 2.9606678294683500 ]
Q”(X(5)) = X(54)X(143) ∩ X(195)X(25043)
= (SB+SC) *(2*R^2-SA-SW) *(2*SB+R^2)*(2*SC+R^2)*(S^2+SA*SB)*(S^2+SA*SC) : : (barys)
= on lines: {54, 143}, {195, 25043}
= [ -46.5519801106257100, 82.5055035321866500, -31.9930009893175800 ]
César Lozada
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