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