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X(30), X(232), X(250), X(520), X(523), X(1113), X(1114), X(1304), X(2966)

X(32230) = isogonal conjugate of X(2972) on the lines {30,250}, {523,1304}

Geometric properties :

The trilinear polar L(P) of any P on the Kiepert hyperbola (K) is perpendicular to the Euler line.

The circles centered on L(P) and orthogonal to the circumcircle (O) form a pencil and pass through two (not necessarily real) points W1, W2 on the Euler line. These points are obviously inverse in (O) and lie on a same circum-conic C(P) passing through X(250).

When P traverses (K), L(P) and C(P) meet at two points M1, M2 which lie on K1097, a nK with root the barycentric square X(23582) of X(648). X(23582) is also the trilinear pole of the line passing through X(107), X(110), X(648), etc.

Walsmith examples :

• P = X(4) gives W1, W2 = X(5000), X(5001) and C(P) passes through X(6). M1, M2 are X(232), X(523).

• P = X(76) gives W1, W2 = X(5002), X(5003) and C(P) passes through X(69).

• P = X(83) gives W1, W2 = X(5004), X(5005) and C(P) passes through X(95).