HYACINTHOS 25491

[Antreas P. Hatzipolakis]:

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

Denote:

A", B", C" = the reflections of P in BC, CA, AB, resp.

A*B*C* = the orthic triangle of A"B"C"

Which is the locus of P such that A'B'C', A*B*C* are perspetive?

I lies on the locus.

 

[César Lozada]:

Locus = a degree-9-excentral circumcurve q9 through ETCs 1, 187 (equation below).

 

If Z(P) is the perspector:

 

Z( X(187) ) = X(2482)

 

For P=I, A’B’C’ and A*B*C* are homothetic, with B*C*/B’C’ = 2*r/R.

 

Z( I ) = midpoint of X(1) and X(3585)

= (b+c)*a^3+(b^2+c^2)*a^2-(b^2- c^2)*(b-c)*a-(b^2-c^2)^2 : : (barycentrics)

= 2*X(1125)-X(5267) = X(2646)-3*X(4870) = 3*X(3584)-X(11010) = X(3916)-2*X(4999)

= On lines:

{1,4}, {2,46}, {3,1770}, {5,65}, {7,90}, {8,6871}, {10,908}, {11,113}, {12,517}, {19,5747}, {20,3612}, {21,36}, {30,2646}, {35,411}, {40,498}, {55,6985}, {56,3560}, {57,499}, {72,2886}, {80,7548}, {115,2653}, {124,1845}, {140,1155}, {142,3624}, {165,6988}, {191,5745}, {235,1905}, {238,1780}, {284,1839}, {354,496}, {355,2099}, {376,4333}, {377,997}, {381,1837}, {386,3914}, {431,1829}, {442,960}, {474,5880}, {484,6684}, {486,2362}, {495,3057}, {519,5086}, {527,6763}, {551,4311}, {553,1776}, {595,3011}, {631,3474}, {758,6734}, {938,6870}, {952,11011}, {962,3085}, {999,10404}, {1001,7742}, {1111,3674}, {1156,5557}, {1158,6833}, {1159,3851}, {1193,3120}, {1210,3671}, {1319,5901}, {1329,3753}, {1385,7354}, {1388,9657}, {1420,4317}, {1452,3542}, {1454,6862}, {1470,7702}, {1482,5252}, {1532,7686}, {1538,5806}, {1565,4059}, {1697,10056}, {1698,2093}, {1708,6832}, {1709,6847}, {1717,3100}, {1723,5746}, {1727,6888}, {1728,6846}, {1738,3216}, {1756,4357}, {1768,6705}, {1788,3090}, {1892,11399}, {1940,7551}, {2051,4424}, {2098,3656}, {2475,4511}, {2800,8068}, {3091,10826}, {3136,10974}, {3146,4305}, {3149,11507}, {3179,5243}, {3304,11373}, {3306,10200}, {3333,4654}, {3336,3911}, {3339,6855}, {3340,5587}, {3428,5812}, {3434,3811}, {3555,3813}, {3576,4299}, {3579,5432}, {3584,11010}, {3601,4302}, {3614,9956}, {3616,4293}, {3634,5445}, {3635,7972}, {3670,8229}, {3683,6675}, {3687,4647}, {3697,9710}, {3698,3820}, {3702,3936}, {3720,4303}, {3746,10624}, {3754,3814}, {3755,5312}, {3812,4187}, {3816,5439}, {3822,3878}, {3841,10176}, {3850,12019}, {3868,10916}, {3899,5837}, {3916,4999}, {3925,5044}, {3931,5718}, {3947,4301}, {4002,9711}, {4047,5742}, {4294,5703}, {4297,10483}, {4298,5563}, {4309,9580}, {4640,7483}, {4679,11108}, {4847,5904}, {4848,6874}, {4867,6737}, {5010,6876}, {5045,7743}, {5083,5533}, {5123,10107}, {5173,5777}, {5218,6361}, {5250,10198}, {5274,11036}, {5328,11024}, {5398,7299}, {5425,6738}, {5433,11230}, {5506,6666}, {5542,10394}, {5657,10588}, {5690,10592}, {5722,10896}, {5726,11531}, {5730,5794}, {5763,7957}, {5905,10527}, {6001,6831}, {6866,9581}, {6875,7280}, {6911,11509}, {6982,7982}, {6990,10395}, {7284,10586}, {7680,10523}, {7965,11018}, {8069,11496}, {9596,9620}, {9597,9619}, {9655,10246}, {10042,11372}, {10057,10698}, {10264,11670}, {10265,11571}, {10679,11501}, {10883,11019}

= midpoint of X(1) and X(3585)

= reflection of X(i) in X(j) for these (i,j): (3916,4999), (5267,1125), (10039,12)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1,4,10572), (1,1699,1479), (1,3583,950), (1,5270,10106), (1,9612,1478), (4,3485,1), (4,3487,10393), (226,946,1), (497,3487,1), (1058,3475,1), (5290,11522,1)

= [ 0.536973414289351, 0.40318948355607, 3.113699417465861 ]

 

César Lozada

 

PD: q9 trilinear equation

 

q9 = q0+CyclicSum[ qc ]   where:

 

q0 = 18*(b^2-c^2)*(a^2-b^2)*(a^2-c^ 2)*(-a^2+b^2+c^2)*(a^2-b^2+c^ 2)*(a^2+b^2-c^2)*a*b*c*u^3*v^ 3*w^3

 

qc = v*w*((-a^2+b^2+c^2)*((a^2-b^2+ 3*c^2)*b^3*v^5-(a^2+3*b^2-c^2) *c^3*w^5)*a^4*b^2*c^2*v*w+4*( b^2-c^2)*(-a^2+b^2+c^2)*a^5*b^ 3*c^3*u^7+((a^8-a^6*b^2-3*a^4* b^4+a^4*b^2*c^2-6*a^4*c^4+5*a^ 2*b^6-2*a^2*b^4*c^2-3*a^2*b^2* c^4+8*a^2*c^6-2*b^8+b^6*c^2+ 13*b^4*c^4+3*b^2*c^6-3*c^8)*b* v^3-(a^8-a^6*c^2-6*a^4*b^4+a^ 4*b^2*c^2-3*a^4*c^4+8*a^2*b^6- 3*a^2*b^4*c^2-2*a^2*b^2*c^4+5* a^2*c^6-3*b^8+3*b^6*c^2+13*b^ 4*c^4+b^2*c^6-2*c^8)*c*w^3)*a^ 4*b*c*v^2*w^2+(-(3*a^8+3*a^6* b^2-13*a^6*c^2-8*a^4*b^4-12*a^ 4*b^2*c^2+12*a^4*c^4-5*a^2*b^ 6+a^2*b^4*c^2+a^2*b^2*c^4+3*a^ 2*c^6+7*b^8-16*b^6*c^2+6*b^4* c^4+8*b^2*c^6-5*c^8)*c*v+(3*a^ 8-13*a^6*b^2+3*a^6*c^2+12*a^4* b^4-12*a^4*b^2*c^2-8*a^4*c^4+ 3*a^2*b^6+a^2*b^4*c^2+a^2*b^2* c^4-5*a^2*c^6-5*b^8+8*b^6*c^2+ 6*b^4*c^4-16*b^2*c^6+7*c^8)*b* w)*a^2*b^2*c^2*u^6-((2*a^10-8* a^8*b^2-3*a^8*c^2+12*a^6*b^4+ 9*a^6*b^2*c^2-4*a^6*c^4-8*a^4* b^6-7*a^4*b^4*c^2+4*a^4*b^2*c^ 4+10*a^4*c^6+2*a^2*b^8-a^2*b^ 6*c^2+8*a^2*b^4*c^4-11*a^2*b^ 2*c^6-6*a^2*c^8+2*b^8*c^2-12* b^6*c^4-5*b^4*c^6+6*b^2*c^8+c^ 10)*b*v-(2*a^10-3*a^8*b^2-8*a^ 8*c^2-4*a^6*b^4+9*a^6*b^2*c^2+ 12*a^6*c^4+10*a^4*b^6+4*a^4*b^ 4*c^2-7*a^4*b^2*c^4-8*a^4*c^6- 6*a^2*b^8-11*a^2*b^6*c^2+8*a^ 2*b^4*c^4-a^2*b^2*c^6+2*a^2*c^ 8+b^10+6*b^8*c^2-5*b^6*c^4-12* b^4*c^6+2*b^2*c^8)*c*w)*a^4*v^ 3*w^3+(-(8*a^10-a^8*b^2-21*a^ 8*c^2-40*a^6*b^4-4*a^6*b^2*c^ 2+6*a^6*c^4+12*a^4*b^6-12*a^4* b^4*c^2-12*a^4*b^2*c^4+28*a^4* c^6+24*a^2*b^8-36*a^2*b^6*c^2+ 2*a^2*b^4*c^4+40*a^2*b^2*c^6- 30*a^2*c^8-3*b^10+5*b^8*c^2-6* b^6*c^4+18*b^4*c^6-23*b^2*c^8+ 9*c^10)*c^2*v^2+(8*a^10-21*a^ 8*b^2-a^8*c^2+6*a^6*b^4-4*a^6* b^2*c^2-40*a^6*c^4+28*a^4*b^6- 12*a^4*b^4*c^2-12*a^4*b^2*c^4+ 12*a^4*c^6-30*a^2*b^8+40*a^2* b^6*c^2+2*a^2*b^4*c^4-36*a^2* b^2*c^6+24*a^2*c^8+9*b^10-23* b^8*c^2+18*b^6*c^4-6*b^4*c^6+ 5*b^2*c^8-3*c^10)*b^2*w^2)*a* b*c*u^5-(b^2-c^2)*(26*a^8-34* a^6*b^2-34*a^6*c^2-21*a^4*b^4+ 30*a^4*b^2*c^2-21*a^4*c^4+40* a^2*b^6-16*a^2*b^4*c^2-16*a^2* b^2*c^4+40*a^2*c^6-11*b^8+20* b^6*c^2-18*b^4*c^4+20*b^2*c^6- 11*c^8)*a*b^2*c^2*u^5*v*w-(b^ 2-c^2)*(4*a^10-14*a^8*b^2-14* a^8*c^2+16*a^6*b^4+31*a^6*b^2* c^2+16*a^6*c^4-4*a^4*b^6-3*a^ 4*b^4*c^2-3*a^4*b^2*c^4-4*a^4* c^6-4*a^2*b^8-31*a^2*b^6*c^2+ 26*a^2*b^4*c^4-31*a^2*b^2*c^6- 4*a^2*c^8+2*b^10+17*b^8*c^2- 51*b^6*c^4-51*b^4*c^6+17*b^2* c^8+2*c^10)*a^3*u*v^3*w^3+(-( 8*a^12+18*a^10*b^2-26*a^10*c^ 2-41*a^8*b^4-57*a^8*b^2*c^2+7* a^8*c^4-38*a^6*b^6+37*a^6*b^4* c^2+16*a^6*b^2*c^4+53*a^6*c^6+ 83*a^4*b^8-15*a^4*b^6*c^2-56* a^4*b^4*c^4+57*a^4*b^2*c^6-61* a^4*c^8-26*a^2*b^10+47*a^2*b^ 8*c^2-6*a^2*b^6*c^4-8*a^2*b^4* c^6-24*a^2*b^2*c^8+17*a^2*c^ 10-4*b^12+14*b^10*c^2-14*b^8* c^4-4*b^6*c^6+16*b^4*c^8-10*b^ 2*c^10+2*c^12)*c*v+(8*a^12-26* a^10*b^2+18*a^10*c^2+7*a^8*b^ 4-57*a^8*b^2*c^2-41*a^8*c^4+ 53*a^6*b^6+16*a^6*b^4*c^2+37* a^6*b^2*c^4-38*a^6*c^6-61*a^4* b^8+57*a^4*b^6*c^2-56*a^4*b^4* c^4-15*a^4*b^2*c^6+83*a^4*c^8+ 17*a^2*b^10-24*a^2*b^8*c^2-8* a^2*b^6*c^4-6*a^2*b^4*c^6+47* a^2*b^2*c^8-26*a^2*c^10+2*b^ 12-10*b^10*c^2+16*b^8*c^4-4*b^ 6*c^6-14*b^4*c^8+14*b^2*c^10- 4*c^12)*b*w)*b*c*u^4*v*w)