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quadrilateral of 4 orthocenters in a cyclic pentagon is square iff criterion

Source: 2016 Bulgaria JBMO TST 2.2

August 11, 2020
geometrysquareorthocenterCyclicpentagon

Problem Statement

The vertices of the pentagon ABCDEABCDE are on a circle, and the points H1,H2,H3,H4H_1, H_2, H_3,H_4 are the orthocenters of the triangles ABC,ABE,ACD,ADEABC, ABE, ACD, ADE respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if BE∥CDBE \parallel CD and the distance between them is BE+CD2\frac{BE + CD}{2}.
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