HYACINTHOS 26416

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

Bc, Cb = the orthogonal projections of B, C on PC, PB, resp.

Ca, Ac = the orthogonal projections of C, A on PA, PC, resp.

Ab, Ba = the orthogonal projections of A, B on PB, PA, resp.

La, Lb, Lc = the perpendicular bisectors of BcCb, CaAc, AbBa, resp 

 

L1, L2, L3 = the perpendicular bisectors of AbAc, BcBa, CaCb, resp.

 

Which is the locus of P such that:

1. La, Lb, Lc are concurrent?
2. L1, L2, L3 are concurrent?

[César Lozada]

 

1)      Locus = {circumcircle} \/ {Line-at-infinity}\/{McCay cubic K003}

If P lies on K003, the point of concurrence Qa(P)=complement(isogonal(P))

ETC pairs (P, Qa=La/\Lb/\Lc): (1, 10), (3, 3), (4, 5), (1075, complement(13855)), (1745, complement(3362)), (3362, complement(1745)), (13855, complement(1075))

 

Qa(1075) = complement of X(13855)
= (16*R^4+8*(SA-SW)*R^2-S^2-2*SA ^2+SW^2)*(2*S^4+(8*(-2*SA+SW)* R^2+(SA+SW)*(3*SA-2*SW))*S^2+( -SW^2+16*R^4)*(SA-SW)*SA) : : (barys)

= on line (2,13855)

= complement of X(13855)

= [ 3.292518089942654, 1.95143249522704, 0.770049020622899 ]

 

Qa(1745) = complement of X(3362)

= ((b+c)*a^5+b*c*a^4-2*(b^3+c^3) *a^3+(b^4-c^4)*(b-c)*a-(b^2-c^ 2)^2*b*c)*((b-c)^2*a^5+b*c*(b+ c)*a^4-2*(b^3-c^3)*(b-c)*a^3+( b^4-c^4)*(b^2-c^2)*a-(b^2-c^2) ^2*(b+c)*b*c) :: (barys)

= On lines: {2,3362}, {1210,1785}

= complement of X(3362)

= [ 1.009648979402880, 0.12594011680483, 3.087483333625859 ]

 

Qa(3362) = complement of X(1745)

= (b-c)^2*a^5-b*c*(b+c)*a^4-2*(b ^3-c^3)*(b-c)*a^3+(b^4-c^4)*(b ^2-c^2)*a+(b^2-c^2)^2*(b+c)*b* c : : (barys)

= on lines: {2,1745}, {3,10}, {29,58}, {57,1940}, {1413,8808}

= reflection of X(10571) in X(1125) , {3741,12514}, {6708,9940}

= complement of X(1745)

= [ 5.689807678830224, 4.67566209173709, -2.222397433370633 ]

 

Qa(13855) = complement of X(1075)

= SA*(SB+SC)*(16*R^4-8*(2*SA+SW) *R^2+S^2+(SA+SW)^2+SA^2) : : (barys)

= X(3)+2*X(8798)

= On lines: {2,1075}, {3,64}, {4,2972}, {394,2055}, {417,11459}, {418,7999}, {852,11412}, {1216,6638}, {3090,13409}

= complement of X(1075)

= {X(6509), X(11793)}-Harmonic conjugate of X(3)

= [ 2.902212807952342, 1.62810130149050, 1.174034592589876 ]

 

2)      Locus={Line-at-infinit y}\/{Gibert’s Q030}

ETC pairs (P, Q1=L1/\L2/\L3/\): (1, 946), (4,5), (13, 13), (14,14), (74,10264), (80,10265)

 

César Lozada