HYACINTHOS 27201

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P, Q two isogonal conjugate points and ApBpCp. AqBqCq the pedal triangles of P, Q, resp.

 

Denote:

A', A'p, A'q  = the orthogonal projections of A, Ap, Aq on PQ. resp.

B', B'p, B'q  = the orthogonal projections of B, Bp, Bq on PQ resp.

C', C'p, C'q  = the orthogonal projections of C, Cp, Cq on PQ. resp.

Conjecture:

The circumcircles of A'AqA'p, B'BqB'p, C'CqC'p and ApBpCp = circumcircle of AqBqCq, are concurrent.

The circumcircles of A'ApA'q, B'BpB'q, C'CpC'q and ApBpCp = circumcircle of AqBqCq, are concurrent

 

[César Lozada]:

 

APH

[Conjecture:

The circumcircles of A'AqA'p, B'BqB'p, C'CqC'p and ApBpCp = circumcircle of AqBqCq, are concurrent at Z1

The circumcircles of A'ApA'q, B'BpB'q, C'CpC'q and ApBpCp = circumcircle of AqBqCq, are concurrent at Z2 ]

Confirmed!.

 

For P=u:v:w (trilinears):

 

Z1(u,v,w) = u*(2*a^3*b*c*(-a^2+b^2+c^2)*v^ 3*w^5+2*a^3*b*c*(-a^2+b^2+c^2) *v^5*w^3+a*(-a^2+b^2+c^2)*(2* a^4-b^4+2*b^2*c^2-c^4-a^2*b^2- c^2*a^2)*u^2*v^4*w^2+a*(-a^2+ b^2+c^2)*(2*a^4-b^4+2*b^2*c^2- c^4-a^2*b^2-c^2*a^2)*u^2*v^2* w^4-2*c*a^2*(-2*b^4+a^2*b^2+b^ 2*c^2-2*c^2*a^2+c^4+a^4)*u*v^ 2*w^5+c*a^2*(-5*b^4+a^4+4*b^2* c^2+c^4-2*c^2*a^2+4*a^2*b^2)* v^4*u*w^3+8*a*b*c*(2*a^4-b^4+ 2*b^2*c^2-c^4-a^2*b^2-c^2*a^2) *u^2*v^3*w^3+c*(11*a^4*b^2-12* a^2*b^4-3*b^2*c^4-b^6+c^6+2*a^ 6+3*b^4*c^2+12*c^2*a^2*b^2-3* a^4*c^2)*v^2*u^3*w^3-2*b*a^2*( b^4+b^2*c^2-2*c^4+c^2*a^2+a^4- 2*a^2*b^2)*u*v^5*w^2+b*(b^6-c^ 6-3*b^4*c^2-3*a^4*b^2+2*a^6+ 11*a^4*c^2+3*b^2*c^4-12*a^2*c^ 4+12*c^2*a^2*b^2)*u^3*v^3*w^2- 6*a^2*b^2*c*(a^2-b^2+c^2)*u^5* w^3-2*a^2*b^2*c*(a^2-b^2+c^2)* u^3*w^5-6*a^2*b*c^2*(a^2+b^2- c^2)*u^5*v^3-2*a^2*b*c^2*(a^2+ b^2-c^2)*u^3*v^5+a*(26*c^2*a^ 2*b^2+4*c^6+3*a^6-4*b^4*c^2-4* b^2*c^4-2*a^4*c^2-5*a^2*b^4-5* a^2*c^4-2*a^4*b^2+4*b^6)*u^4* v^2*w^2-3*b*a^2*(a^2-b^2+c^2)* (-a^2+b^2+c^2)*u^5*v*w^2-4*a^ 3*b*c*(-a^2+b^2+c^2)*u^6*v*w- 3*c*a^2*(a^2+b^2-c^2)*(-a^2+b^ 2+c^2)*u^5*v^2*w-4*c^2*b^2*a^ 3*u^6*w^2-4*c^2*b^2*a^3*u^6*v^ 2+b*a^2*(-5*c^4+b^4+4*b^2*c^2+ 4*c^2*a^2-2*a^2*b^2+a^4)*u*v^ 3*w^4-2*a*b*c*(2*a^4-b^4+2*b^ 2*c^2-c^4-a^2*b^2-c^2*a^2)*u^ 4*v^3*w-2*a*b*c*(2*a^4-b^4+2* b^2*c^2-c^4-a^2*b^2-c^2*a^2)* u^2*v*w^5-2*a*b*c*(2*a^4-b^4+ 2*b^2*c^2-c^4-a^2*b^2-c^2*a^2) *u^2*v^5*w-2*a*b*c*(2*a^4-b^4+ 2*b^2*c^2-c^4-a^2*b^2-c^2*a^2) *u^4*v*w^3+a^3*(b^4+c^4-2*a^2* b^2+6*b^2*c^2+a^4-2*c^2*a^2)* v^4*w^4-2*a*c^2*(-2*b^2*c^2-2* c^2*a^2+a^4+c^4+4*a^2*b^2+b^4) *u^4*v^4-2*a*b^2*(a^4+c^4+4*c^ 2*a^2+b^4-2*b^2*c^2-2*a^2*b^2) *u^4*w^4-c*(c^6+2*a^2*c^4-b^6- 7*a^4*c^2-3*b^2*c^4+6*c^2*a^2* b^2+4*a^6-8*a^2*b^4+3*b^4*c^2+ 5*a^4*b^2)*u^3*v^4*w-b*(5*a^4* c^2+3*b^2*c^4+b^6+2*a^2*b^4-7* a^4*b^2-3*b^4*c^2-8*a^2*c^4+4* a^6+6*c^2*a^2*b^2-c^6)*u^3*v* w^4)*(-2*a*b*c*u^2*v^2-2*a*b* c*u^2*w^2+4*a*b*c*v^2*w^2+a*(- a^2+b^2+c^2)*v^3*w+a*(-a^2+b^ 2+c^2)*v*w^3-b*(a^2-b^2+c^2)* u*w^3-c*(a^2+b^2-c^2)*u*v^3+( a^2-b^2+c^2)*b*u*v^2*w+(a^2+b^ 2-c^2)*c*u*v*w^2-2*a*(-a^2+b^ 2+c^2)*v*u^2*w)*(-c*(a^2+b^2- c^2)*u^2*v+b*(a^2-b^2+c^2)*u^ 2*w-2*a*u*b*v^2*c+c*(a^2+b^2- c^2)*v*w^2-b*(a^2-b^2+c^2)*v^ 2*w+2*w^2*a*u*b*c)^2/a : :

 

Z2(u,v,w)= Z1(1/u, 1/v, 1/w)

 

Example:

 

Z1( X(3) ) = complement of X(1304)

= SA*(S^2-3*SB*SC)*(2*S^2+SA^2+ 2*SB*SC-SW^2) *(5*S^2-24*R^2*SA+5*SA^2-4*SB* SC) : : (barys)

= on the nine-points-circle and on these lines: {2, 1304}, {4, 2693}, {30, 133}, {113, 2072}, {114, 5159}, {115, 6587}, {122, 523}, {128, 6760}, {131, 10257}, {132, 858}, {136, 3154}, {1560, 3163}, {1650, 3258}, {2972, 6070}, {3150, 5099}, {3548, 15454}, {5627, 15404}, {12079, 15526}, {13573, 15384}

= midpoint of X(i) and X(j) for these {i,j}: {4, 2693}, {13573, 15384}

= complement of X(1304)

= orthoptic circle of Steiner inellipse-inverse-of X(2697)

= [ 3.6705157383351010, 2.9425438080090500, -0.0905654182534709 ]

 

Z2( X(3) ) = polar circle-inverse-of X(10420)

= SB*SC*(6*R^2-SA-SW)*(2*S^2+SA^ 2+2*SB*SC-SW^2)*((14*R^2-3*SW) *S^2-(6*R^2-SW)*SA^2) : : (barys)

= on the nine-points-circle and on these lines: {4, 10420}, {113, 10151}, {115, 6753}, {131, 403}, {135, 523}

= polar circle-inverse-of X(10420)

= [ 3.7122788571904470, -0.4982877431011221, 2.2722734468895660 ]

 

César Lozada