Source — Euclid's Elements, encoded as an rrxiv paper

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III.5 Proposition III.5

If two circles cut one another, they will not have the same centre.

Proof

Let circles

Γ_{1}

and

Γ_{2}

meet at points

A

and

B

. Suppose they share centre

E

. Then

E A

is a radius of

Γ_{1}

and also of

Γ_{2}

; the two circles thus have the same centre and the same radius, so they coincide — contradicting their meeting at only two points (or in general, being two distinct circles).

lines 74–74 in main.tex