How can I convince my friend her circles aren’t round?

BILL— Everyone from time to time needs a visit from the Geometry Police. We see circles which are flat sided, ones which go into corners, and ones which begin and end at the wrong place. We draw our students elaborate diagrams of “tangent points” replete with eight-by-ten colour glossy photographs with circles and arrows and a paragraph on the back of each one. We distribute the Cones of Shame in the arena. Sometimes none of it does any good because they have convinced themselves that their circles are already round enough.

Then it’s time for the ultimate slap upside the head. You will need the following props: the rider’s horse all tacked up and ready to be ridden normally, four traffic cones, a flat area large enough that you can mark the four corners of a 20 meter square with excess room on the outside all around it, and a lunge line.

Now mark off the length of exactly 10 meters on the line. You will stand at the exact center of the square and pivot on that point, never being drawn off it. Take the opposite end of the line and — without fastening it to the horse or the rider – place it on the tread of the stirrup under the rider’s inside boot. She should press down on it enough with the ball of her foot to hold the end of the line in place.

Have her simply ride a succession of correct 20 meter circles inscribed in that square. Retain your spot at all costs, keep the line the same length, and continue facing her as she turns around you.
The rider does the steering. If her horse falls in, the line droops and becomes slack. If her horse bulges out, the line will pull her inside leg away from the horse’s side. It’s an immediate (and mildly embarrassing) way to discover how unround supposedly “round” circles can be!