Equation 1:
(0  h) ² + (21  k) ² = r ²
h ² + 441  42k + k ² = r ²
441  42k = r ²  h ²  k ²
Equation 2:
(13.5  h) ² + (13.5  k) ² = r ²
182.25  27h + h ² + 182.25  27k + k ² = r ²
364.5  27h  27k = r ²  h ²  k ²
Equation 3:
(22  h) ² + (0  k) ² = r ²
484  44h + h ² + k ² = r ²
484  44h = r ²  h ²  k ²
The right sides of the equations are equal, therefore all the equations are equal to one another.
The following examples could have equated Equation 2 and Equation 3,
and solved for h first rather than k.
Left Side of Equation 1 = Left Side of Equation 2
441  42k = 364.5  27h  27k
15k = 76.5  27h>
k = (76.5  27h) / 15
Left Side of Equation 1 = Left Side of Equation 3
441  42k = 484  44h
 42k = 43  44h
k = (43  44h) /  42
Both equations equal k, therefore the equations are equal to each other.
(76.5  27h) / 15 = (43  44h) /  42
 42(76.5  27h) = 15(43  44h)
3213 + 1134h =  645 + 660h
474h =  3858
h =  8.13924
Substituting the value of h in either equation expressed in terms of k:
k = (76.5  27 ×  8.13924) / 15 =  9.55063
or
k = (43  44 ×  8.13924) /  42 =  9.55063
h and k may be substituted in any of the original equations to determine the radius of the circle.
Equation 1:
(0  ( 8.13924)) ² + (21  ( 9.55063)) ² = r ²
8.13924 ² + 30.55063 ² = r ²
66.24723 + 933.34099 = r ²
999.58822 = r ²
Taking the square root of both sides of the equation:
r = 31.61627
Equation 2:
(13.5  ( 8.13924)) ² + (13.5  ( 9.55063)) ² = r ²
21.63924 ² + 23.05063 ² = r ²
468.25671 + 531.33154 = r ²
999.58825 = r ²
Taking the square root of both sides of the equation:
r = 31.61627
Equation 3:
(22  ( 8.13924)) ² + (0  ( 9.55063)) ² = r ²
30.13924 ² + 9.55603 ² = r ²
908.37379 + 91.21453 = r ²
999.58832 = r ²
Taking the square root of both sides of the equation:
r = 31.61627
