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In a cyclic quadrilateral ABCD, the four vertices taken three at a time are considered as a triangle and the orthocenter of each triangle is found. The resulting four orthocenters also form a cyclic quadrilateral whose center is the reflection of the circumcenter of the original quadrilateral in the anti-center, (the point where the maltitudes intersect). You may move the four original vertices to see the resulting orthic quadrilateral.

Note that the original quadrilateral and the orthic quadrilateral share maltitudes and thus have a common anti-center

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