[alogo] Simson related locus

From a point P draw parallels to the altitudes of the triangle ABC. Project to each parallel the vertex belonging to the parallel altitude. The three projections G, I, J are on a line, exactly when P is on the circumcircle of the antiparallel triangle DEF of ABC, resulting by drawing parallels to the sides from the opposite vertices of ABC.

[0_0] [0_1]
[1_0] [1_1]

Obvious proof since, for example, projection point G on PG is always on side FD of the antiparallel triangle. Thus the condition of collinearity becomes equivalent to the condition of collinearity of projections of P on the sides of DEF.

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