HYACINTHOS 28387

[César Lozada]:

 

Let ABC be a triangle, P a point and A’B’C’, A”B”C” the circumcevian, circum-anticevian(*) triangles of P. Then the vertex triangle of A’B’C’ and A”B”C” degenerates to a point.

 

If P=x : y : z (barys), then the degenerate vertex triangle is:

 

Q(P) = (a^4*y^2*z^2-(b^4*z^2+c^4*y^2)*x^2)*a^2 : :

 

If P lies on the circumcircle ᴦ of ABC, then Q(P)=P.

 

Note: This construction leads to the same results than TCC perspectors described in the preamble just before ETC X(1631).

 

ETC pairs (P ∉ ᴦ, Q): (1,3), (2,22), (3,1498), (4,24), (5,1601), (6,6), (7,1602), (8,1603), (9,1604), (13, 1605), (14, 1606), (19,1609), (21,1610), (25,1611), (28,1612), (31,1613), (54,1614), (55,1615), (56,1616), (57,1617), (58,595), (59,1618), (63,1619), (64,1620), (81,1621), (83,1078), (84,1622), (86,23374), (88,1623), (162,1624), (163,1625), (174,1626), (188,2933), (251,1627), (254,1628), (259,198), (266,56), (275,1629), (284,1630), (365,55), (366,1631), (508,23852), (509,1486), (648,1632), (651,1633), (662,1634), (2153,11142), (2154,11141), (6727,23846), (6733,23845), (14085,101), (14089,99), (18297,23849), (18753,2176)

 

Some others:

Q( X(15) ) = X(6)X(16459) ∩ X(15)X(1495)

= (SB+SC)*(3*(6*R^2+SA)*S^2-sqrt(3)*(5*S^2-3*(6*R^2-3*SA+2*SW)*SA)*S-9*SB*SC*SW) : : (barys)

= on lines: {6, 16459}, {15, 1495}, {22, 2923}

= isogonal conjugate of the cyclocevian conjugate of X(19778)

= [ -5.4667260740576400, -8.1980759495302300, 11.8393598657626800 ]

 

Q( X(16) ) = X(6)X(16460) ∩ X(16)X(1495)

= (SB+SC)*(3*(6*R^2+SA)*S^2-sqrt(3)*(5*S^2+3*(6*R^2-3*SA+2*SW)*SA)*S-9*SB*SC*SW) : : (barys)

= on lines: {6, 16460}, {16, 1495}, {22, 2924}

= isogonal conjugate of the cyclocevian conjugate of X(19779)

= [ 2.4571711953508650, -1.1587468922405610, 3.3087948556043670 ]

 

César Lozada

 

(*): The circum-anticevian triangle of a point P is the triangle whose vertices are the second intersections of the sides of the anticevian triangle of P with the circumcircle of the reference triangle ABC.