HYACINTHOS 25506

[Antreas P. Hatzipolakis]:

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

Denote:

Ab, Ac = the orthogonal projections of A' on AC,AB, resp.

R1 = the radical axis of the circles (Ab, AbA'), (Ac, AcA').

Similarly R2, R3.

A*B*C* = the triangle bounded by R1,R2,R3.

1. Which is the locus of P such that R1, R2, R3 are concurrent?

2. Which is the locus of P such that ABC, A*B*C* are parallelogic?

I lies on the locus. The parallelogic center (ABC, A*B*C*) is the internal center of similitude of (I) and (O) = X55

 

[César Lozada]:

 

1)      Locus1 = { Line-at-infinity} \/ {circum-quartic q4 through ETCs 3, 4, 523} (q4 equation below)

ETC pairs (P, point of concurrence): (3, 141), (4, 52)

 

2)      Locus2 = { Line-at-infinity} \/ {Darboux cubic } \/ { same q4}

Some parallelogic centers Za =(A->A*) and Z*(A*->A) for P on Darboux cubic:

 

Za( X(20) ) = X(25)

 

Z*( X(20) ) = a*((b^2+c^2)*a^8-2*(b^4+c^4)* a^6+4*b^2*c^2*(b^2+c^2)*a^4+2* (b^8+c^8-2*(b^4+b^2*c^2+c^4)* b^2*c^2)*a^2-(b^4-c^4)^2*(b^2+ c^2)) : : (trilinears)

= (4*cos(2*A)+cos(4*A)-1)*cos(B- C)-(3*cos(A)+cos(3*A))*cos(2*( B-C))+9*cos(A)-cos(3*A) : : (trilinears)

= On lines: {20,2979}, {22,1495}, {51,858}, {161,1350}, {185,1993}, {394,1619}, {511,1370}, {1216,11414}, {1843,7391}, {2071,5012}, {3060,7396}, {3819,7493}, {5447,9715}, {7667,9967}, {7998,10565}, {10625,11750}

= [ 12.079979060451090, 10.44599125456820, -9.166550568386416 ]

 

Za( X(40) ) = X(56)

 

Z*( X(40) ) = midpoint of X(1479) and X(5904)

= (b+c)*a^5-(b^2+c^2)*a^4-(b+c)* (2*b^2-3*b*c+2*c^2)*a^3+(2*b^ 4+2*c^4+b*c*(b+c)^2)*a^2+(b+c) *(b^4+c^4-3*b*c*(b^2+c^2))*a-( b+c)*(b^2-c^2)*(b^3-c^3) : : (trilinears)

= csc(A/2)^2*((3*sin(A/2)-4*sin( 3*A/2)+sin(5*A/2))*cos((B-C)/ 2)+(cos(A)-cos(2*A))*cos(B-C)- cos(A)+1) : : (trilinears)

= on lines: {63,3678}, {72,515}, {78,2801}, {144,3648}, {200,7992}, {329,1479}, {518,10392}, {758,3436}, {908,3825}, {1898,5853}, {2802,3632}, {2975,10176}, {3421,5693}, {3585,5176}, {3680,9951}, {3681,4882}, {3868,11678}, {3927,11499}, {4847,5777}, {5442,5744}, {5883,11681}, {6001,6736}, {6763,10090}

= midpoint of X(1479) and X(5904)

= reflection of X(3874) in X(3825)

= [ 6.755440989247031, 4.04994514117107, -2.281039533940377 ]

 

César Lozada

q4 (trilinears):

CyclicSum[ a*v*w*(2*((a^8-(3*b^2+2*c^2)* a^6+3*(b^2-c^2)*b^2*a^4-(b^2- c^2)*(b^4+5*b^2*c^2+2*c^4)*a^ 2+(b^2-c^2)^3*c^2)*v^2-(a^8-( 2*b^2+3*c^2)*a^6-3*(b^2-c^2)* c^2*a^4+(b^2-c^2)*(2*b^4+5*b^ 2*c^2+c^4)*a^2-(b^2-c^2)^3*b^ 2)*w^2)*b^2*c^2+4*(b^2-c^2)*( 3*a^4-4*(b^2+c^2)*a^2+(b^2-c^ 2)^2)*b^3*c^3*v*w-(b^2-c^2)*( a^10-5*(b^2+c^2)*a^8+2*(5*b^4+ 6*b^2*c^2+5*c^4)*a^6-2*(b^2+c^ 2)*(5*b^4-2*b^2*c^2+5*c^4)*a^ 4+(5*b^4-14*b^2*c^2+5*c^4)*(b^ 2+c^2)^2*a^2-(b^4-c^4)*(b^2-c^ 2)^3)*u^2) ] = 0