HYACINTHOS 26900

[Antreas P. Hatzipolakis]:

Hyacinthos 26893-NPC, radical axes, reflections, locus

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

(Ma), (Mb), (Mc) = the circles with diameters ANa, BNb, CNc, resp.

R1 = the radical axis of (Mb), (Mc)
R2 = the radical axis of (Mc), (Ma)
R3 = the radical axis of (Ma), (Mb)

Which is the locus of P such that the reflections of R1, R2, R3 in BC, CA, AB, resp. are concurrent?

I lies on the locus. The point of concurrence is the I.

[César Lozada]:

 

Locus = {circumcircle} ∪ {q7: a degree-7 excentral-circumcurve through the vertices of triangles ABC-X3 REFLECTIONS, EXCENTRAL, ORTHIC and ETC’s  1, 3, 4}

 

q7 = ∑ [ y*z*((c^2*(a^8-4*c^2*a^6-(6*b^ 4-b^2*c^2-6*c^4)*a^4+2*(4*b^6- 2*c^6-b^2*c^2*(b^2+2*c^2))*a^2 -(b^2-c^2)*(3*b^6+c^6-2*b^2*c^ 2*(b^2-2*c^2)))*y-b^2*(a^8-4* b^2*a^6+(6*b^4+b^2*c^2-6*c^4)* a^4-2*(2*b^6-4*c^6+b^2*c^2*(2* b^2+c^2))*a^2+(b^2-c^2)*(b^6+ 3*c^6+2*b^2*c^2*(2*b^2-c^2)))* z)*x^4-(b^2-c^2)*(a^8-4*(b^2+ c^2)*a^6+(6*b^4+7*b^2*c^2+6*c^ 4)*a^4-4*(b^4+b^2*c^2+c^4)*(b^ 2+c^2)*a^2+(b^4+7*b^2*c^2+c^4) *(b^2-c^2)^2)*z*y*x^3-2*a^2*( a^2-b^2-c^2)*(a^4-3*(b^2+c^2)* a^2+2*(b^2-c^2)^2)*(b^2-c^2)*z ^2*y^2*x+a^6*(a^2-b^2-c^2)*(a^ 2-b^2+c^2)*z^2*y^3-a^6*(a^2+b^ 2-c^2)*(a^2-b^2-c^2)*z^3*y^2) ] = 0 (barycentrics)

 

The curve q7 seems to be nice (see attached figure). Maybe Angel would like to put an eye on it.

 

* For P on q7, ETC pairs (P, Q(P)=point of concurrence): (1, 1), (3, 5), (4, 5)

 

* For P on the circumcircle, R1, R2, R3 are perpendicular to BC, CA, AB and their reflections in the sidelines are themselves. In this case Q(P) = reflection in N of the antipode of P in the circumcircle. Moreover, the locus of Q(P) is the circle H(R), through ETC’s 265, 6033, 6321, 7728, 10738, 10739, 10740, 10741, 10742, 10743, 10744, 10745, 10746, 10747, 10748, 10749, 12918, 13556, 14980.

 

ETC-pairs (P, Q(P)):

 

(74,265), (98,6321), (99,6033), (100,10742), (101,10741), (102,10747), (103,10739), (104,10738), (109,10740), (110,7728), (112,12918), (1292,10743), (1293,10744), (1294,10745), (1296,10748), (1297,10749), (1300,13556)

 

Some others:

Q( X(105) ) = reflection of X(1292) in X(5)

= a^8-2*(b+c)*a^7+(2*b^2-b*c+2*c ^2)*a^6-(b+c)*(2*b^2-7*b*c+2*c ^2)*a^5-b*c*(7*b^2-4*b*c+7*c^2 )*a^4+(b+c)*(2*b^4+2*c^4+b*c*( 3*b^2-8*b*c+3*c^2))*a^3-2*(b+c )*(b^2-c^2)*(b^3-c^3)*a^2+2*(b ^2-c^2)*(b-c)*(b^4+c^4+b*c*(b- c)^2)*a-(b^4-c^4)*(b^2-c^2)*( b-c)^2 : : (barys)

= 3*X(3)-4*X(6714), 2*X(120)-3*X(381), 3*X(5511)-2*X(6714)

= on lines: {3, 5511}, {4, 10743}, {5, 1292}, {30, 105}, {120, 381}, {265, 2775}, {528, 3830}, {1358, 1479}, {1478, 3021}, {2788, 6321}, {2795, 6033}, {2809, 10741}, {2814, 10747}, {2820, 10739}, {2826, 10738}, {2835, 10740}, {2836, 7728}, {2838, 12918}, {3627, 10729}, {3845, 10712}, {9519, 10744}, {9520, 10745}, {9521, 10746}, {9522, 10748}, {9523, 10749}

= reflection of X(i) in X(j) for these (i,j): (3, 5511), (1292, 5), (10712, 3845), (10729, 3627), (10743, 4)

= [ -9.7650531181757380, -3.0765962522825580, 10.2775633264918700 ]

 

 

Q( X(106) ) = reflection of X(1293) in X(5)

= a^7-2*(b+c)*a^6-3*(b^2-3*b*c+c ^2)*a^5+3*b*c*(b+c)*a^4-10*b^2 *c^2*a^3+(b+c)*(3*b^4+3*c^4-2* b*c*(3*b^2-4*b*c+3*c^2))*a^2+( b^2-c^2)^2*(2*b^2-9*b*c+2*c^2) *a-(b^2-c^2)^2*(b+c)*(b^2-3*b* c+c^2) : : (barys)

= 3*X(3)-4*X(6715), 2*X(121)-3*X(381), 3*X(5510)-2*X(6715)

= on lines: {3, 5510}, {4, 10744}, {5, 1293}, {30, 106}, {121, 381}, {265, 2776}, {1357, 1479}, {1478, 6018}, {2789, 6321}, {2796, 6033}, {2802, 10742}, {2810, 10741}, {2815, 10747}, {2821, 10739}, {2827, 10738}, {2841, 10740}, {2842, 7728}, {2844, 12918}, {3627, 10730}, {3845, 10713}, {9519, 10743}, {9524, 10745}, {9525, 10746}, {9526, 10748}, {9527, 10749}

= reflection of X(i) in X(j) for these (i,j): (3, 5510), (1293, 5), (10713, 3845), (10730, 3627), (10744, 4)

= [ -12.0488339167987500, -6.2738893703547760, 13.5451266229832700 ]

 

César Lozada