Proof of Congruencies

We will only show one congruence proof, triangle DHC is congruent to triangle DPC. The other congruencies can be obtained similarly.

Since these two triangles already have a right angle at D in common and side DC in common, we will try to prove their congruence using ASA. So, we need to show that angle BCP is congruent to angle DCH.

Since triangle AFH is similar to triangle CDH, angle FAH is congruent to angle DCH. We will label these angles with 1's for illustration.

But, angle FAH and angle BCP both subtend arc BP. So, angle FAH must also be congruent to angle BCP. So, we will label angle BCP with a 1 as well.

Thus we have angle BCP congruent to angle DCH.

So, we can conclude that triangle DHC is congruent to triangle DPC by ASA.

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