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 or 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