From geometry to coordinates to circle regression

The SAT asks about circle equations, coordinates, distance and circle theorems. It never asks for circle regression by name. Regression is your shortcut for the hard ones, where the problem gives you enough exact facts to pin down one circle, but finding the center by hand, with perpendicular bisectors, would take too long.

Here's what that looks like.

Fast Desmos solution, in four moves:

1. Pick coordinates. Put $\overline{AB}$ on the $x$-axis with its midpoint at the origin, so $A=(-8,0)$ and $B=(8,0)$. The rectangle is $4$ tall, and $E$ is halfway across the top, so $E=(0,4)$.
2. Fit the circle. Put the three points in a Desmos table and type x_1^2+y_1^2~ux_1+vy_1+w. Desmos shows $u=0$, $v=-12$ and $w=64$.
3. Read the circle. Halve $u$ and $v$ to get the center, $(0,-6)$. Then $r^2=w+0^2+6^2=64+36=100$. Section 2 shows where these two rules come from, so for now, follow along.
4. Check it. Each point's squared distance from the center should be $100$. $A$ and $B$ each give $8^2+6^2=100$, and $E$ gives $0^2+10^2=100$. So the radius is $\boxed{10}$.

Notice that the center landed below the rectangle. That's hard to guess from a sketch, but the fit doesn't need a guess.

Those four moves are the whole method, in that order. Desmos handles the fit, but picking the points and checking the exact answer are on you. Desmos can't invent a fact the problem didn't give, it can't stand in for a theorem, and a rounded decimal on the screen doesn't prove a value is exact.

You know the center-radius equation:

You could type that into Desmos and fit $h$, $k$ and $r$ directly, and it can work. But look where the unknowns sit: $h$ and $k$ are inside squares, so that's a nonlinear regression. There's a friendlier way to write the same circle. Expand the squares and move every term with $h$, $k$ or $r$ to the right:

Treat $2h$, $2k$ and $r^2-h^2-k^2$ as three single unknown numbers. Each one is only multiplied by $x$, by $y$ or by nothing. That's what linear in its parameters means: no unknown is squared or multiplied by another unknown.

This is the form we'll use every time, for a practical reason. Whatever the question asks for, you get it from these three numbers with the same short routine: halve two of them for the center, add the center's squares to the third for $r^2$, and flip signs for the expanded equation.

Put the points in a Desmos table as x_1 and y_1, then type:

x_1^2+y_1^2~ux_1+vy_1+w

Line it up with the expanded equation, term by term. $u$ sits where $2h$ was, $v$ where $2k$ was, and $w$ where $r^2-h^2-k^2$ was. So $u$ is twice the center's $x$-coordinate, $v$ is twice its $y$-coordinate, and adding $h^2+k^2$ back to $w$ leaves $r^2$:

In the calculators here, q is a short name for $r^2$, typed as q=w+h^2+k^2.

Sometimes the question wants the equation with everything on the left and $0$ on the right:

The model has $ux+vy+w$ on the right, so subtract all three terms from both sides:

Now match the two equations term by term. Each term changed sides, so each sign flipped:

Three exact points that aren't on one line allow only one circle, so this fit lands on that circle every time. It's an exact fit, not a best fit through scattered data. The only catch is that Desmos shows rounded decimals, which is why its answer still needs your check.