Proof
Suppose two circles touch at two points , . By III.11
(internal) or III.12 (external), both and lie on the line
joining the centres. Thus this line cuts each circle in two
points, making it a diameter of each. But then is a chord of
each circle equal in length to the diameter — so and are
antipodal points on each circle, and both circles share centre and
diameter, contradicting III.5/III.6.
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Full neighborhood
Depends on (4)
- III.5Proposition III.5If two circles cut one another, they will not have the same centre.
- III.6Proposition III.6If two circles touch one another, they will not have the same centre.
- III.11Proposition III.11If two circles touch one another internally, and their centres be taken, the straight line joining their centres, if…
- III.12Proposition III.12If two circles touch one another externally, the straight line joining their centres will pass through the point of…
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