Concyclic Quadrilateral

The points at the feet of the altitudes

create a quadrilateral. This quadrilateral is similar to the quadrilateral created by the reflections of E over

from the problem. Our goal is to prove the smaller

to be cyclic, and then extending this to the larger one

To prove the smaller one to be cyclic I will be using a theorem that states that: "If the opposite angles of a quadrilateral are supplementary then it is cyclic" Lets prove this for

by showing that:

Remember we want to show that

Lets further break this down by showing that

and

are complementary as well as

and

Consider the

. This is easily shown to be cyclic because

by definition of altitudes. NOTE: This is also true for

and

. There is an important angle relation that this will give us.

This is because for any 2 points that subtend the same chord (such that those points arent on the chord), their respective formed angles will be equivalent Note we can do the same thing for

and

Since

and

are complementary (due to an orthodiagonal

and

are as well! The same follows for

and

. We have verified that

and that

is cyclic! This is easily extended to

. Consider

, it has midpoints L and M for sides

and

, which means

. Repeat for the rest of the sides. All expected angles are equal. Opposite angles are supplementary and

is a set of concyclic points :D

Concyclic Quadrilateral

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