Proposition 13
Theorem
If two circles are tangent, then they have exactly one point in common.
Commentary
1. Given two circles, let one pass through points A, B, C, and the other pass through points D, E, F.
2. Let these two circles be tangent at two points K and H.
3. Then these two points must be the same.
4. Equivalently, two circles cannot be tangent to each other at two different points. Euclid's original statement of this proposition is phrased in terms of this impossibility.
2. Let these two circles be tangent at two points K and H.
3. Then these two points must be the same.
4. Equivalently, two circles cannot be tangent to each other at two different points. Euclid's original statement of this proposition is phrased in terms of this impossibility.
Original statement
κύκλος κύκλου οὐκ ἐϕάπτϵται κατὰ πλϵίονα σημϵῖα ἢ καθ᾽ ἕν, ἐάν τϵ ἐντὸς ἐάν τϵ ἐκτὸς ἐϕάπτηται.
English translation
A circle does not touch a circle at more points than one, whether it touches it internally or externally.
