HYACINTHOS 27233

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, A'B'C' the pedal triangle of H and L = the Euler line.

Denote:

La, Lb, Lc = the reflections of L in BC, CA, AB, resp.

Ha, Hb, Hc = the orthogonal projections of H on La, Lb, Lc, resp.
Oa, Ob, Oc = the orthogonal projections of O on La, Lb, Lc, resp.

The circumcircles of A'HaOa, B'HbOb, C'HcOc are coaxial.

It seems the intersection points of the circles are always real. 

Which are they?

 

[César Lozada]:

 

X(113) & X(1986)

 

The radical trace is X(11557) and the radical axis is the tripolar of:

 

T = isotomic conjugate of X(15421)

= SB*SC*(SA-SB)*(SA-SC)*(2*SA+ SB+SC-6*R^2) : : (barys)

= on lines: {2, 216}, {99, 112}, {250, 4226}, {378, 6795}, {476, 1304}, {523, 4230}, {687, 15421}, {11061, 13200}

= isotomic conjugate of X(15421)

= polar conjugate of X(15328)

= trilinear pole of the line {113, 403}

= barycentric product X(i)*X(j) for these {i,j}: {99, 403}, {113, 16077}, {264, 15329}, {648, 3580}, {811, 1725}, {892, 12828}, {3003, 6331}, {6528, 13754}

= barycentric quotient X(i)/X(j) for these (i,j): (4, 15328), (107, 1300), (110, 5504), (112, 14910), (113, 9033), (186, 15470), (250, 10420), (403, 523), (476, 12028), (648, 2986), (686, 3269), (1304, 10419), (1725, 656), (1986, 526), (2315, 822), (3003, 647), (3580, 525), (4240, 15454), (6334, 15526), (12824, 9517), (12826, 2850), (12828, 690), (13754, 520), (14264, 14380), (15329, 3)

= trilinear product X(i)*X(j) for these {i,j}: {92, 15329}, {162, 3580}, {403, 662}, {648, 1725}, {811, 3003}, {823, 13754}, {2315, 6528}

= trilinear quotient X(i)/X(j) for these (i,j): (92, 15328), (113, 2631), (162, 14910), (403, 661), (662, 5504), (811, 2986), (823, 1300), (1725, 647), (1986, 2624), (3003, 810), (3580, 656), (6334, 2632), (12828, 2642), (13754, 822), (15329, 48)

= [ -0.3551414992398414, 0.4224235046916784, 3.5121289013854530 ]

 

César Lozada