SOLUTION: A circle has its center at (6,7) and passes through (1,4). A second circle is tangent to the first circle at (1,4) and has the same area. What are coordinates of the center of th

Circles are all the same shape, so if they have equal area, they are
exactly alike.  So we draw a circle with center (6,7) that goes through
(1,4), and another circle just like it toching it at (1,4), like this.



We want to find the center of the circle on the left.

Let's start by drawing a verticle and horizontal path from the center (6,7)
to the point (1,4) like the red arrow below.  That means we'll go down 3 units
to get on the same level with (1,4): 



and now we'll go left 5 units and we're at (1,4). 



Now we'll do the same thing, this time starting at (1,4), and go down 3 units, 
and go left 5 units, like this:




And now we just observe what point we ended up on, which is (-4,1).

Answer: the center of the circle on the left is (-4,1).

Edwin