Mathlover's way is rather difficult.  Substituting those points into 

 and getting the system of three equations:



It's complicated but if you simplify all those and substitute, you'll
get h, k, and r, as she has shown above.

Here's an easier way but it's just about as long:  
Draw the three points and connect two pairs
of them.



We find the equations of the perpendicular bisectors of each chord.

The slope of the shorter chord, using the slope formula is 1/3

The midpoint of the shorter chord, using the midpoint formula is (5/2,7/2),

So the perpendicular bisector of the short chord has slope which is the negative
reciprocal of 1/3 which is -3, and it goes through (5/2,7/2)

So, using the point-slope equation of a line, and simplifying, the perpendicular
bisector of the shorter chord has equation y = -3x+11

Doing the exact same thing with the longer chord, we find that its slope is -2
and its midpoint is (6,0).  

So the perpendicular bisector of the longer chord has slope which is the
negative reciprocal of -2 which is 1/2, and it goes through (6,0).

Using the point-slope equation of a line, and simplifying, the perpendicular
bisector of the longer chord has equation .

These perpendicular bisectors of the two chords are plotted in red below:



The two perpendicular bisectors (in red) must intersect at the center of the
circle, so we solve the system of equations:



and get their point of intersection as (4,-1), which is the center of the
circle.

We could use the distance formula to find the radius.  However it is not
necessary in this case because one of the given point (4,4), just happens
to be exactly 5 units above the center (4,-1), so we know that the radius
is 5.

So, since h=4,   k=-1, and  r=5, the standard equation   

 becomes



.



Edwin