"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle ABCD and outside ACK, which is absurd. Therefore a circle does not touch a circle externally at more points than"
There's only one little detail which i'm not sure of. We are trying to prove that circles which touch one another will only touch at one point. Fine. I understood the first part which treats of a circle in another one. It's only the case where one circle touches another one from the outside. By using proposition 2 of book 3, we prove that the line AC will be inside both of circles since the two points are on each circumference of the two circles. Now, this is where I get lost. We say that "but it (line AC) fell within the circle ABCD and outside ACK" and we prove this by using definition 3 of book 3 (Circles are said to touch one another which meet one another but do not cut one another.) In other words, this definition says that circles which touch another do not cut one another. In our situation, we have two circles which touch one another and are not supposed to cut one another. This is where I don't understand, how does this justify this : "but it fell within the circle ABCD and outside ACK." How do we get that conclusion from the definition?
Thank you!
http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII13.html
Given the line AC,it lies inside the circles ACK and ACBD since we had assumed that the circles intersect at A and C.
But,AC lies inside ABCD and outside AKC,and AKC contains AC(i.e.the circle AKC contains and does not contain AC at the same time),which is absurd.