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How would I find the x and y coordinates of the center of a circle if I am given the coordinates for 3 random points on the circumference of said circle? I also know the radius of the circle. I am 11 years old and I haven't taken trigonometry or geometry yet, so I am unsure how to approach this problem. Circle Diagram

Goomba
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    You probably only need two, actually. Find the line connecting, say, $A$ and $B$. Find its midpoint, $M$. Find the line perpendicular to that line through $M$. Find where it intersects the circle, say, $P,Q$. Find the midpoint of $P$ and $Q$. Dunno how viable this is in practice but it is how you could do it with compass-and-straightedge construction. – PrincessEev Aug 17 '23 at 13:47
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    @PrincessEev with two points and the length of the radius, you may get two possible centres. Just draw the circles radius $r$ round $A$ and round $B$ and see where they intersect. – Henry Aug 17 '23 at 14:06
  • Oh, I see, hence the third point to ensure a unique circle. – PrincessEev Aug 17 '23 at 14:16
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    Lots of useful information on this question https://math.stackexchange.com/questions/213658/get-the-equation-of-a-circle-when-given-3-points – Brian Lai Aug 17 '23 at 14:18
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    Three non collinear points uniquely determine a circle. So adding n extra radius is not necessary, and it might be wrong. See https://en.wikipedia.org/wiki/Circumcircle – Andrei Aug 17 '23 at 14:45

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We have three points: $A(x_a, y_a), B(x_b, y_b), C(x_c, y_c)$. The center $O(x,y)$ is equidistant from all three points. Hence, $$\begin{cases} (x-x_a)^2+(y-y_a)^2=(x-x_b)^2+(y-y_b)^2 \\ (x-x_c)^2+(y-y_c)^2=(x-x_b)^2+(y-y_b)^2 \end{cases} $$ or $$\begin{cases} 2x(x_b-x_a)+2y(y_b-y_a)=x_b^2+y_b^2-x_a^2-y_a^2 \\ 2x(x_b-x_c)+2y(y_b-y_c)=x_b^2+y_b^2-x_c^2-y_c^2 \end{cases} $$ Solve the system to find the center.

Vasili
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The circle center is crossed by all perpendiculars of straight line segments that run through the centers of these line segments. So one can construct 3 line segments from the 3 points, construct their mid points, and construct lines orthogonal to the line segments; their common intersection is the circle center.

R. J. Mathar
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If you were to connect all 3 given points with line segments to form an inscribed $\triangle$, the $\triangle$'s circumcenter would coincide with the ⭕️'s center point. To find the circumcenter, we need to find the $\perp$ bisectors for at least 2 of the $\triangle$'s 3 sides. The point of intersection for these lines is the circumcenter.

Ryan Wilson
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