# Equation of a circle: the Mathematical Ninja

“Four points,” said the student. “On a circle.”

The Mathematical Ninja nodded, impatiently.

“$A(-5,5)$, $B(1,5)$, $C(-3,3)$ and $D(3,3)$,” he read from the book, for the third time.

A slight crack of a smile. It may have been a snarl. You can never tell with the Mathematical Ninja.

“I’m terribly sorry, I don’t know where to start.”

In his head, the Mathematical Ninja arranged the student’s large intestine in a circle passing through the four points; he was aware, though, that Ofsted didn’t consider that to be outstanding teaching.

(If YOU don’t know where to start, you could start by reading this article about where to start when you don’t know where to start.)

“Picture,” coughed the Mathematical Ninja.

The student painstakingly drew out his axes, while the Mathematical Ninja sharpened *his* axes. “There!” said the student, at last. “Now what?

“Wellouch,” said the Mathematical Ninja, irritatedly running his finger along a blade and carefully reattaching it, “what does it look like?”

“Er… what’s the equation of a circle, again?”

The axes did a very good job of pinning the student’s clothes to the spinning wooden wheel; no student was *physically* harmed in the writing of this article.

### Listen up, sunshine, I’m only going to say this once

“The picture, my little chickadee, is a quadrilateral. A trapezium, if you look at it from this perspective.” He span the wheel around again and stopped it with the student pointing at a jaunty angle of $\frac{\pi}{4}$ radians.

The student wisely thought better of asking how that helped.

“The centre is equidistant from all of the points, so it’s on the perpendicular bisector of any pair of them.”

“So it’s on the… can you put me straight so I can tell which axis is which? Thanks. The $x$-axis?”

“Correct.”

“Oh, and the $x$-coordinate is midway between $-5$ and $1$… so $x=2$?

“Right.”

“And then is it Pythagoras to find the radius?”

“You did know, after all.” That was a snarl. Definitely a snarl.

“So, the centre’s at $(2,0)$ and the radius is $\sqrt{34}$, so the equation is $(x-2)^2 + y^2 = 34$.”

“An equation,” said the Mathematical Ninja. “AN equation.”