Proposition 10
Theorem
If two circles have more than two points in common, they must coincide.
Commentary
1. Given two circles, one passing through A, B, C, D, and the other through B, C, D, E, so that they have three points in common.
2. Then these two circles must coincide.
3. Euclid stated the proposition in a negative way, while this site uses the contrapositive of his statement.
4. Book 3 Propositions 5 and 6 talk about when two circles have one or two points in common, while this proposition covers the case when two circles have more than two points in common.
5. This proposition can be summed up by saying three non-collinear points uniquely define a circle.
2. Then these two circles must coincide.
3. Euclid stated the proposition in a negative way, while this site uses the contrapositive of his statement.
4. Book 3 Propositions 5 and 6 talk about when two circles have one or two points in common, while this proposition covers the case when two circles have more than two points in common.
5. This proposition can be summed up by saying three non-collinear points uniquely define a circle.
Original statement
κύκλος κύκλον οὐ τέμνϵι κατὰ πλϵίονα σημϵῖα ἢ δύο.
English translation
A circle does not cut a circle at more points than two.
