Proposition III.5 — If two circles cut one another, they will not have the same …

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).

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Depends on (2)

I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…
III.1Definition III.1Equal circles are those whose diameters are equal, or whose radii are equal.

Required by (dependents) (3)

III.6Proposition III.6If two circles touch one another, they will not have the same centre.
III.10Proposition III.10A circle does not cut a circle at more points than two.
III.13Proposition III.13A circle does not touch a circle at more points than one, whether it touch it internally or externally.

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