How Many?!

Geometry Level 3

Given a triangle $ABC$ , drop perpendiculars from $A$ to $BC$ , $B$ to $CA$ and $C$ to $AB$ , and let the feet of these perpendiculars be $D$ , $E$ and $F$ . It is well known that the three perpendiculars concur (meet) at a point $H$ , the orthocentre of $ABC$ .

In general, how many distinct cyclic quadrilaterals are there with vertices that are among $A, B, C, D, E, F$ and $H$ ?

Note : This is not an original problem (although I like it very much)

The answer is 6.

1 solution

Arjen Vreugdenhil
Dec 9, 2017

There are $\boxed{6}$ such quadrilaterals.

Three include two vertices and the two feet on either side, e.g. $ADEB$ ;
Three include one vertex, the two adjacent feet, and the orthocenter, e.g. $AEHF$ .

This accounts for six of the 35 combinations that can be made with $ABCDEFH$ . Of the remaining 29 combinations,

24 would result in a degenerate quadrilateral because three points would be colinear;
2 are $ABCH$ and $DEFH$ , which would necessarily be concave;
3 would consist of the three feet and one vertex, e.g. $DEAF$ , forming a convex quadrilateral which however is not cyclic.

How Many?!

The answer is 6.

1 solution

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