HYACINTHOS 24776

[Antreas P. Hatzipolakis]:

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

Denote:

Oa,Ob,Oc = the circumcenters of AB'C', BC'A',CA'B' resp. ( = midpoints of AP,BP,CP, resp.)

O1, O2, O3 = the reflections of Oa, Ob, Oc in PA', PB', PC', resp.

For P = N, the ABC and O1O2O3 are bilogic ( =  perspective and orthologic). .

Which is the locus of P such that ABC, O1O2O3 are:
 
1. perspective

2. orthologic
?

[César Lozada]:

1)      Locus = the cubic with barycentric equation:

CyclicSum[ SA*((4*SA*SB+SB^2+S^2)*y-(4* SC*SA+SC^2+S^2)*z)*x^2 ] - (b^2-c^2)*(c^2-a^2)*(a^2-b^2) *x*y*z=0

This cubic passes throughs ETC’s: 4, 5, 83

 

Perspectors Z(P):

Z(H) = H; Z(N) = X(3519); Z(X(83)) = X(7768)

 

2)      Locus = all the plane. For P = u : v : w (trilinears) the orthologic centers A->O1=Za and O1->A = Zo are:

Za(P) =  a/(SA*a^3*u-(-SW*SA+SA^2+2*S^ 2)*(b*v+c*w)) : :

Zo(P) = (a*(SA^2-SW*SA-2*S^2)*u+SB*SC* (b*v+c*w))/a : :

 

ETC pairs (P,Za(P)):

(1,3417), (2,3431), (3,3), (4,4), (5,54), (6,3425), (10,947), (52,8884), (64,5879), (68,254), (69,7612), (76,3406), (113,250), (114,249), (115,2065), (119,59), (125,10419), (141,5481), (155,24), (195,3432), (265,523), (355,1), (381,6), (399,3447), (546,1173), (946,58), (1351,25), (1352,2), (1482,56), (1657,3532), (2072,5504), (2080,3455), (2095,1436), (2888,3459), (3095,32), (3448,1138), (3574,1166), (3652,35), (3656,2163), (3818,262), (3830,3426), (3843,3527), (5403,1343), (5404,1342), (5446,1179), (5480,251), (5562,96), (5611,3438), (5613,16), (5615,3439), (5617,15), (5777,943), (5779,55), (5805,57), (5887,21), (5891,95), (6033,511), (6246,1168), (6248,83), (6265,36), (6287,39), (6288,5), (6289,371), (6290,372), (6321,512), (6841,1175), (7574,67), (7680,3449), (7681,3450), (7682,3451), (7683,3453), (7686,961), (7697,182), (7728,30), (8724,187), (8905,8883), (8906,6193), (9927,847), (9967,1799), (9970,23), (10113,5627), (10297,895), (10525,5553), (20749,3433), (20750,3435)

 

ETC pairs (P,Zo(P)):

(1,5882), (2,3), (3,550), (4,4), (5,140), (6,8550), (20,1657), (40,5493), (51,389), (54,10619), (186,10295), (262,39), (376,20), (381,5), (403,468), (428,6756), (546,3850), (547,3530), (549,548), (568,6102), (631,3522), (1598,7715), (1699,946), (1853,6247), (2394,5489), (2888,3519), (3060,52), (3089,3517), (3090,3523), (3091,1656), (3146,5073), (3153,7574), (3524,376), (3529,5059), (3541,3516), (3542,3515), (3543,382), (3544,3533), (3545,2), (3576,4297), (3817,1125), (3830,3627), (3832,3851), (3839,381), (3843,3858), (3845,546), (3855,5056), (5054,8703), (5055,549), (5064,1595), (5066,3628), (5067,10299), (5071,631), (5093,1353), (5102,3629), (5476,575), (5587,10), (5603,1), (5627,6070), (5640,9730), (5654,1147), (5655,5609), (5656,1498), (5657,40), (5658,1490), (5790,5690), (5817,9), (5886,1385), (5890,185), (5891,1216), (5902,5884), (5927,5777), (5943,9729), (6194,9821), (6530,1990), (7552,7488), (7565,5576), (7576,3575), (7710,8721), (7714,7487), (7775,7764), (7967,944), (7988,10165), (9166,6055), (9752,3053), (9753,32), (9754,5206), (9779,5886), (10157,5044), (10170,5447), (10175,6684), (10201,1658), (10247,1483), (10304,3534), (10516,141), (10519,1350), (10606,5894)

 

Notes:

Zo(P) is the reflection in P of the midpoint of {P, H} (this last one always lies on the np circle), i.e. PZo(p)= -1/2*PH.

When P moves on the circumcircle of P, Zo(P) moves on the circle with center=X(550) and radius=3*R/2. No ETC center X(n) lies on this circle (for  n<=10682). Some points on this circle:

 

Zo( X(74) ) = reflection of X(125) in X(74)

= (8*cos(2*A)+10)*cos(B-C)-cos( A)*cos(2*(B-C))-15*cos(A)-2* cos(3*A) : : (trilinears)

= reflection of X(i) in X(j) for these (i,j): (125,74), (146,5972),…

= On lines: {3,541}, {4,74}, {20,542}, {30,6070}, {64,67}, {110,3522}, {113,140}, {146,3523}, {185,1205}, {265,5073}, {548,5609}, {550,5562}, {690,5489},

 

Zo( X(98) ) = reflection of X(115) in X(98)

= SA*(3*S^2-2*SW^2)*(SA-2*SW)- SW^2*(S^2+2*SA^2) : : (barycentrics)

= Circumcircle-inverse-of-X( 5621)

= midpoint of X(i) and X(j) for these {i,j}: {98,9862}, {99,5984},…

= reflection of X(i) in X(j) for these (i,j): (115,98), (147,620),…

= On lines: {3,67}, {4,32}, {5,6055}, {20,543}, {39,5477}, {99,3522}, {114,140}, {125,5191}, {147,620}, {148,5059}, {187,1503}, {446,9420}, {550,2782}, {574,6776}, {626,9863}, {631,6054}, {671,3146}, {690,5489}, {754,5999},…

 

Zo( X(99) ) = reflection of X(114) in X(99)

= (12*S^2-SW^2)*SA^2-(9*S^2-SW^ 2)*SW*SA+(6*S^2-SW^2)*S^2 : : (barycentrics)

= reflection of X(i) in X(j) for these (i,j): (114,99), (148,6036),…

= On lines: {3,543}, {4,99}, {5,2482}, {20,542}, {98,3522}, {115,140}, {147,5059}, {148,3523}, {382,8724}, {550,2782}, {620,1656}, {631,671}, {1657,2794},…

= [ 7.996403885935437, 0.99033257092664, -0.735598322242728 ]

 

Zo( X(100) ) = reflection of X(119) in X(100)

= 4*a^6*(-b-c+a)-(7*b^2-6*b*c+7* c^2)*a^5+(b+c)*(7*b^2-2*b*c+7* c^2)*a^4+2*(b^2+b*c+c^2)*(b^2- 4*b*c+c^2)*a^3-(b+c)^2*(b-c)^ 2*(2*(b+c)*a^2-a*(b^2+c^2)+(b^ 2-c^2)*(b-c)) : : (barycentrics)

= reflection of X(i) in X(j) for these (i,j): ((119,100), (149,6713),…

= On lines: {3,528}, {4,100}, {5,6174}, {11,35}, {30,5537}, {40,550}, {104,3522}, {149,3523}, {153,5059}, {517,10609},…

= [ 6.741063907926889, 1.03543687569204, -0.187436697230216 ]

 

César Lozada