Get the equation of a circle when given 3 points Get the equation of a circle through the points $(1,1), (2,4), (5,3) $.
I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?

Follow these steps:


*

*Consider the general equation for a circle as $(x-x_c)^2+(y-y_c)^2 - r^2 = 0$

*Plug in the three points to create three quadratic equations
$$ (1-x_c)^2+(1-y_c)^2 - r^2 = 0 $$
$$ (2-x_c)^2+(4-y_c)^2 - r^2 = 0 $$
$$ (5-x_c)^2+(3-y_c)^2 - r^2 = 0 $$

*Subtract the first from the second, and the first from the third to create two linear equations
$$ -2 x_c -6 (y_c-3)=0 $$
$$ (y_c+7)-6 x_c = 0 $$

*Solve for the center as 
$$ (x_c,y_c) = (3,2) $$

*Plug the values for the center in any of the three quadratic equations above (I choose the first) and solve for $r$
$$ (1-3)^2+(1-2)^2-r^2 = 0 $$ 
$$ 5-r^2 = 0 $$
$$ r = \sqrt 5 $$

*Verify result with GeoGebra (optional)



