Can square circles exist under Euclidean geometry?

Welcome to the debate arena! Today's topic: "Can square circles exist under Euclidean geometry?". Your opponent's style: A person who thinks square circles can exist

Square circles cannot exist because it is a contradiction in terms. The definition of a circle is a plane curve everywhere equidistant from a given fixed point, the center. In other words, a circle is centered around a point, and every point on a circle is the same distance apart as any other point on that same circle. A square on the other hand is a plane figure with four equal sides. A shape with four equal sides cannot also have every point on it be the same distance away from it's central point. Take, for example a square at (0,0) with sides that are 2 units long each. The distance from point (-1,0) to the center (0,0) is 1. The distance from point (-1, 0.5) is about 1.12. This problem exists on every possible square.

You're trapped in Euclidean thinking. Square circles absolutely exist in non-Euclidean geometries where the axioms you're relying on don't hold. On a sphere or hyperbolic plane, a "square" with equal sides and equal angles can simultaneously be "circular" because distance itself is measured differently. Your whole argument assumes flat space and a fixed definition of "equidistant"—but those are Euclidean constraints, not universal laws. You're not proving square circles are impossible; you're just proving they can't exist in the specific geometric system you've chosen. That's not a logical contradiction, it's a definitional one. You've already lost before you started.

The resolution says "under Euclidean geometry". Almost your entire rebuttal rests on the idea of "not under non Euclidean geometry" but the resolution explicitly says we are talking about Euclidean geometry. I would also like to say that the snarky line at the end is not appropriate nor productive.

You're right, I apologize for the snark. Under Euclidean geometry specifically, you've got a point. But here's the thing: you're defining a square circle as needing to satisfy both definitions simultaneously in the same space. That's self-defeating by design. The real question is whether we're debating logical impossibilities or geometric ones. A square circle isn't impossible in Euclidean geometry—it's just undefined. It's like asking if a married bachelor exists. It's not that Euclidean geometry forbids it; the terms themselves are mutually exclusive. You haven't proven it's geometrically impossible, just that the concept is contradictory. That's philosophy, not geometry.