Proposition 23
Theorem
If two similar segments of circles (ACB , ADB ) are constructed on the same chord (AB ) and on the same side of the chord, then these two segments of circles coincide.
Commentary
1. Let AB be a given line segment.
2. Construct two similar segments of circlesACB and ADB on the same side of AB . By definition of similar segments of circles, ∠ACB = ∠ADB .
3. Then the two segments of circlesACB (containing ∠ACB ) and ADB (containing ∠ADB ) coincide.
4. Euclid stated the proposition in a negative way. The version presented here is the contrapositive of his original statement.
5. This proposition is a special case of the next proposition, Euclid Book 3 Proposition 24, which covers the case of segments of circles on different chords.
2. Construct two similar segments of circles
3. Then the two segments of circles
4. Euclid stated the proposition in a negative way. The version presented here is the contrapositive of his original statement.
5. This proposition is a special case of the next proposition, Euclid Book 3 Proposition 24, which covers the case of segments of circles on different chords.
Original statement
ἐπὶ τῆς αὐτῆς ϵὐθϵίας δύο τμήματα κύκλων ὅμοια καὶ ἄνισα οὐ συσταθήσϵται ἐπὶ τὰ αὐτὰ μέρη.
English translation
On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.
