HYACINTHOS 26662

 [Amtreas P. Hatzipolakis]:

 

Let ABC be a triangle and P,Q two isogonal conjugate points.

Denote:

(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
(N1), (N2), (N3) = the NPCs of QBC, QCA, QAB, resp.

R1 = the radical axis of (Na), (N1)
R2 = the radical axis of (Nb), (N2)
R3 = the radical axis of (Nc), (N3)

They are concurrent at a point U. [I have posted it : See Hyacinthos 21938 and 21940]
R1 intersects (Na) and (N1) at A* other than the midpoint of BC

R2 intersects (Nb) and (N2) at B* other than the midpoint of CA

R3 intersects (Nc) and (N3) at C* other than the midpoint of AB

The perpendiculars to R1 at A*, R2 at B*, R3 at C* are concurrent at a point  V.

UV is perpendicular to PQ

 

[César Lozada]:

 

 

>The radical axis are concurrent at a point U. [I have posted it : See Hyacinthos 21938 and 21940 ]

> The perpendiculars to R1 at A*, R2 at B*, R3 at C* are concurrent at a point  V.

 

For P=u:v:w (trilinears)

V(P) = ((v^2-w^2)*u*cos(B)*cos(C)+(u^ 2-w^2)*v*cos(B)-(u^2-v^2)*w* cos(C))*((u^2-v^2)*w*cos(B)-( u^2-w^2)*v*cos(C)-(v^2-w^2)*u) : :

 

ETC pairs (P,V):

(3,125), (4,125), (30,6070), (40,2968), (74,6070), (84,2968), (98,6071), (99,6072), (100,6073), (101,6074), (103,14505), (104,6075), (110,1553), (111,6076), (511,6071), (512,6072), (513,6073), (514,6074), (516,14505), (517,6075), (523,1553), (524,6076), (1292,14506), (1293,14507), (1296,6077), (1379,14501), (1380,14502), (1499,6077), (3309,14506), (3667,14507), (6212,1146), (6213,1146)

 

V( X(2) ) =  (SB-SC)^2*(3*SA-2*SW)*(SA^2+2* SB*SC) : : (barycentrics)

= reflection of X(6791) in X(5512)

= On lines: {4, 542}, {1499, 2686}, {1513, 1533}

= [ -3.512564282812110, -3.19591673184997, 7.474405734486103 ]

 

V( X(5) ) =(SB-SC)^2*(S^2-3*SB*SC)*(3*R^2-2*SA)*(SA^2-3*S^2) : :  (barycentrics)

= On lines: {30, 113}, {125, 13152}, {137, 1510}

= [ -8.256800611434413, -9.57273292058971, 14.078772170670060 ]

 

> UV is perpendicular to PQ

 

Confirmed. Their point of intersection is:

Z = ((a^2-b^2+3*c^2)*a*b^2*u*v^6* w^2+(a^2-b^2+c^2)*b^2*c*u^2*v^ 6*w-2*a^2*b^2*c*u^6*w^3+2*a^2* b^2*c*v^6*w^3+2*a^2*b*c^2*u^6* v^3-2*a^2*b*c^2*v^3*w^6+2*a*b* c*(b^2+c^2-a^2)*v^5*u*w^3-2*a* b*c*(b^2+c^2-a^2)*v^5*u^3*w-2* a*b*c*(b^2+c^2-a^2)*v^3*w^5*u+ 2*a*b*c*(b^2+c^2-a^2)*v*w^5*u^ 3-2*a*b*c*(b^2+c^2-a^2)*v*u^5* w^3+2*a*b*c*(b^2+c^2-a^2)*v^3* u^5*w+(b^2-c^2)*a*u*v^4*w^4*( b^2+c^2-a^2)-(b^2-c^2)*a*u^5* v^2*w^2*(b^2+c^2-a^2)-b*a^2*( b^2+c^2-a^2)*w^2*u^6*v+c*a^2*( b^2+c^2-a^2)*v^2*u^6*w+(a^2+b^ 2-c^2)*b*c^2*u^4*v^5+(a^2+3*b^ 2-c^2)*a*c^2*u^5*v^4-(a^2+b^2- c^2)*b*c^2*u^2*v*w^6-(a^2+3*b^ 2-c^2)*a*c^2*u*v^2*w^6+(a^4-2* a^2*b^2-a^2*c^2+b^4-b^2*c^2)* c*u^4*v^4*w+(a^4-a^2*b^2-2*a^ 2*c^2-b^2*c^2+c^4)*b*u^2*v^5* w^2-c*a^2*(b^2+c^2-a^2)*v^4*w^ 5+b*a^2*(b^2+c^2-a^2)*w^4*v^5- (a^4-a^2*b^2-2*a^2*c^2-b^2*c^ 2+c^4)*b*u^4*v*w^4+2*(a^4-a^2* b^2+3*a^2*c^2+b^2*c^2-c^4)*b* u^2*v^3*w^4-2*(a^4-a^2*b^2+3* a^2*c^2+b^2*c^2-c^4)*b*u^4*v^ 3*w^2-2*(a^2*b^2+a^2*c^2-b^4+ 6*b^2*c^2-c^4)*a*u^3*v^4*w^2+ 2*(a^2*b^2+a^2*c^2-b^4+6*b^2* c^2-c^4)*a*u^3*v^2*w^4-(a^4-2* a^2*b^2-a^2*c^2+b^4-b^2*c^2)* c*u^2*v^2*w^5+2*(a^4+3*a^2*b^ 2-a^2*c^2-b^4+b^2*c^2)*c*u^4* v^2*w^3-(a^2-b^2+3*c^2)*a*b^2* u^5*w^4-2*(a^4+3*a^2*b^2-a^2* c^2-b^4+b^2*c^2)*c*u^2*v^4*w^ 3-(a^2-b^2+c^2)*b^2*c*u^4*w^5) *(-(a^2+b^2-c^2)*c*u^2*v+(a^2- b^2+c^2)*b*u^2*w-2*a*b*c*u*v^ 2+2*a*b*c*u*w^2-(a^2-b^2+c^2)* b*v^2*w+(a^2+b^2-c^2)*c*v*w^2) : :  (trilinears)

 

ETC pairs: (3,3154), (4,3154), (30,12079), (74,12079), (101,3234), (110,3233), (514,3234), (523,3233)

 

Z( X(2) ) = (SB-SC)^2*(3*SA-2*SW)*(6*(36* R^2-5*SW)*S^2-2*(9*SA^2-6*SA* SW+4*SW^2)*SW) : : (barycentrics)

= On lines: {2, 6}, {1499, 2686}

= [ -2.055588939224164, 0.99840622696381, 3.898193527497508 ]

 

César Lozada