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X(2), X(4), X(20), X(193)

feet of altitudes

E = SA[(b^2-c^2)^2+a^2(2b^2+2c^2-3a^2)] : : = SA(SCA+SAB-SBC) : : , isoconjugate of X(4)

F = a^2 SA^2 / (SA^2+SBC) : : , on the line X(20)X(193)

K182 answers a question raised by Antreas Hatzipolakis. Consider an isosceles tetrahedron ABCD. Let P be a point in the plane of ABC. It has a "counterpart'' in each of the other three planes. The pedals of these counterparts on ABC form a triangle perspective with ABC if and only if P lies on K182 = pK(X20, X4). (Paul Yiu, 15 nov. 2003)