Geometry

In order to solve this problem, let's analyze somethings about it. 

First, let's note that right triangles $ABC, ABE, ABD, BHD, BEC$ are all 30-60-90 triangles. 

Also, note that right triangles $ABE,ADC$ are 45-45-90 triangles. 

In order to solve the problem, let's draw a line to help find HCA for us. 

Let's draw the altitude of ABC by connecting c to line segment AB. 

This creates a right triangle!

Since we know that

$\angle BAC =60^\circ$

HCA would have to be 30 degrees because of the right triangle.

So we have

$\angle HCA = 30^\circ$

So 30 degrees is our answer. 

Thanks! :)