Math CentralQuandaries & Queries


Question from Carly, a student:

Suppose ABCD is a cyclic quadrilateral, i.e A, B, C, and D are the points on a circle,
given in order going around the circle. Show that if we join each of A, B, C, and D to the orthocentre
of the triangle formed by the other three, then the resulting line segments all intersect in a common midpoint.
Thank you.


First draw a neat figure and label the orthocenters A’, B’, C’, and D’, where A’ is the orthocenter of the triangle BCD that leaves A out, and so on. Can you prove that the sides of the new quadrilateral A’B’C’D’ are parallel to the corresponding sides of the given quadrilateral? (By that I mean A’B’ is parallel to AB, and so on.) If you can, then it shouldn’t be too hard to prove that the segments AA’, BB’, CC’, and DD’ all have a common midpoint.


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