HYACINTHOS 29165

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

A1, B1, C1 = the midpoints of AP, BP, CP, resp
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.

PA' intersects again (Na) at A*
PB' intersects again (Nb) at B*
PC' intersects again (Nc) at C*

 

A*B1 intersects (Nc) again at Ac
A*C1 intersects (Nb) again at Ab

 

B*C1 intersects (Na) again at Ba
B*A1 intersects (Nc) again at Bc

 

C*A1 intersects (Nb) again at Cb

C*B1 intersects (Na) again at Ca

 

A#B#C# = the triangle bounded by AbAc, BcBa, CaCb.

 

Which is the locus of P such that

1. ABC, A#B#C# are perspective?
2, ABC, A#B#C# are orthologic?
3. The perpendicular bisectors of AbAc, BcBa, CaCb are concurrent?
4. The perpendicular bisectors of BcCb, CaAc, AbBa are concurrent ?

 

Special Case: P = I:

AbAc. BcBa, CaCb  are parallels to BC, CA, AB resp (and B1C1, C1A1, A1B1, resp.) passing through A1, B1, C1, resp.
That is, A#B#C# is the antimedial (anticomplementary) triangle of A1B1C1.

 

 

[César Lozada]:

 

1)      Locus = {Linf} ∪  {degree-8 through ETCs 80, 110} ∪ {degree-9 through ETCs 1, 13, 14}

Perspectors Q1(P):

Q1( X(1) ) = X(1125)  (homothetic center)

Q1( X(13) ) = X(17)

Q1( X(14) ) = X(18) 

Q1( X(80) ) = X(10)  (homothetic center, triple perspective)

Q1( X(110) ) = X(5)  (homothetic center, triple perspective)

 

2)      Locus is complicated.

Some triads (P, orthologic centers Q2a, Q2#): (1, 4, 4297), (13, 13, 3), (14, 14, 3) , (74, 265, 550), (80, 1, 10), (110, 1, 5)

 

3)  The entire plane.

Point of concurrence Q3(P) = same Q(P) as in Hyacinthos #28629

 

4)  The entire plane.

For p=x:y:z (barys), the point of concurrence Q4(P) has these elegant coordinates:

Q4(P) = x*(x*((SA+SB)*y^2*(x*SA-SB*y)+(SA+SC)*z^2*(x*SA-SC*z))

           +x*y*z*(2*(S^2+SA^2)*x+(3*S^2-SA*SB)*y+(3*S^2-SA*SC)*z)

           +y*z*((S^2-SB^2)*y^2+2*(S^2+SB*SC)*y*z+(S^2-SC^2)*z^2)) : :

 

If P lies in the infinity then Q4(P) =P

If P lies on the circumcircle of ABC then Q4(P) = N = X(5)

Other ETC pairs (P, Q4(P)): (1,10), (2,16509), (3,5449), (4,5), (5,32551), (6,16511), (13,2), (14,2), (67,16511), (80,10), (265,5449), (671,16509), (1263,32551)

 

Q4( X(11 ) ) = X(5)X(513) ∩ X(10)X(3900)

= (b-c)*(a^5-(2*b^2-b*c+2*c^2)*a^3+(b+c)*b*c*a^2+(b^2-c^2)^2*a-2*(b^2-c^2)*(b-c)*b*c)*(-a+b+c) : : (barys)

= lies on these lines: {5, 513}, {10, 3900}, {522, 3634}, {4885, 6706}, {7412, 16228}

= X(6132)-of-K798i triangle

= center of the circle {X(i), X(j), X(k)} for these {i, j, k}: {10, 946, 20418}, {140, 9955, 9956}

= [ -3.0506044051629300, 5.9879765860192290, 0.9031134939693425 ]

 

Q4( X(15) ) = COMPLEMENT OF X(11127)

= ((b^2+c^2)*a^2-(b^2-c^2)^2)*((-a^2+b^2+c^2)*sqrt(3)+2*S) : : (barys)

= (S^2+SB*SC)*(sqrt(3)*SA+S) : : (barys)

= 3*X(2)+X(19778)

= lies on these lines: {2, 18}, {5, 51}, {95, 302}, {141, 16537}, {184, 5872}, {231, 396}, {298, 340}, {618, 6105}, {621, 19772}, {623, 3580}, {627, 8839}, {850, 20579}, {2041, 11090}, {2042, 11091}, {3129, 5617}, {6117, 14918}, {10601, 11311}, {13350, 13372}, {19295, 23303}

= midpoint of X(11127) and X(19778)

= complement of X(11127)

= isotomic conjugate of the polar conjugate of X(6117)

= {X(2), X(19778)}-harmonic conjugate of X(11127)

= [ 13.9493844583524300, -5.3372793298701280, 0.8975265756548879 ]

 

Q4( X(16) ) = COMPLEMENT OF X(11126)

= ((b^2+c^2)*a^2-(b^2-c^2)^2)*((-a^2+b^2+c^2)*sqrt(3)-2*S) : : (barys)

= (S^2+SB*SC)*(sqrt(3)*SA-S) : : (barys)

= 3*X(2)+X(19779)

= lies on these lines: {2, 17}, {5, 51}, {95, 303}, {141, 16536}, {184, 5873}, {231, 395}, {299, 340}, {619, 6104}, {622, 19773}, {624, 3580}, {628, 8837}, {850, 20578}, {2041, 11091}, {2042, 11090}, {3130, 5613}, {6116, 14918}, {10601, 11312}, {13349, 13372}, {19294, 23302}

= midpoint of X(11126) and X(19779)

= complement of X(11126)

= isotomic conjugate of the polar conjugate of X(6116)

= {X(2), X(19779)}-harmonic conjugate of X(11126)

= [ -0.8930272247953221, 0.2236176355476846, 3.8980186841261300 ]

 

Q4( X(20) ) = X(2)X(31361) ∩ X(4)X(20208)

= (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(4*a^12-5*(b^2+c^2)*a^10-(11*b^4-30*b^2*c^2+11*c^4)*a^8+14*(b^4-c^4)*(b^2-c^2)*a^6+14*(b^2-c^2)^4*a^4-(b^4-c^4)*(b^2-c^2)*(25*b^4-34*b^2*c^2+25*c^4)*a^2+(9*b^4+22*b^2*c^2+9*c^4)*(b^2-c^2)^4) : : (barys)

= (S^2-2*SB*SC)*(7*S^2+16*R^2*(16*R^2-2*SA-7*SW)+8*SA^2-5*SB*SC+12*SW^2) : : (barys)

= lies on these lines: {2, 31361}, {4, 20208}, {5, 11204}, {5893, 20204}, {5895, 20207}

= [ -3.3836825590533060, -4.4050749782326440, 8.2520314171700430 ]

 

César Lozada