PROPOSITION 9.

Magnitudes which have the same ratio to the same are equal to one another ; and magnitudes to which the same has the same ratio are equal.

For let each of the magnitudes A, B have the same ratio to C ;

I say that A is equal to B.

A B C

For, otherwise, each of the magnitudes A, B would not have had the same ratio to C ; [V. 8] but it has ;

therefore A is equal to B.

Again, let C have the same ratio to each of the magnitudes A, B ;

I say that A is equal to B.

For, otherwise, C would not have had the same ratio to each of the magnitudes A, B ; [V. 8] but it has ;

therefore A is equal to B.
Therefore etc. Q. E. D.

Dependency Graph

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Required for

VI.2, VI.3, VI.4, VI.5, VI.6, VI.7, VI.8, VI.9, VI.10, VI.11, VI.12, VI.13, VI.14, VI.15, VI.16, VI.17, VI.18, VI.19, VI.20, VI.22, VI.23, VI.24, VI.25, VI.26, VI.27, VI.28, VI.29, VI.30, VI.31, VI.32, VII.19, VII.24, VII.25, VII.26, VII.27, VII.30, VII.33, VII.34, VII.36, VII.39, VIII.2, VIII.3, VIII.4, VIII.5, VIII.6, VIII.7, VIII.8, VIII.9, VIII.14, VIII.15, VIII.16, VIII.17, VIII.20, VIII.21, VIII.22, VIII.23, VIII.24, VIII.25, VIII.26, VIII.27, IX.1, IX.2, IX.3, IX.4, IX.5, IX.6, IX.8, IX.9, IX.10, IX.12, IX.13, IX.14, IX.15, IX.18, IX.19, IX.32, IX.36, X.6, X.7, X.9, X.10, X.11, X.12, X.13, X.14, X.17, X.18, X.19, X.20, X.21, X.22, X.23, X.24, X.25, X.26, X.27, X.28, X.29, X.30, X.31, X.32, X.33, X.34, X.35, X.36, X.37, X.38, X.39, X.40, X.41, X.42, X.43, X.44, X.45, X.46, X.47, X.48, X.49, X.50, X.51, X.52, X.53, X.54, X.55, X.56, X.57, X.58, X.59, X.60, X.61, X.62, X.63, X.64, X.65, X.66, X.67, X.68, X.69, X.70, X.71, X.72, X.73, X.75, X.76, X.77, X.78, X.79, X.80, X.81, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.92, X.93, X.94, X.95, X.96, X.97, X.98, X.99, X.100, X.101, X.102, X.103, X.104, X.105, X.106, X.107, X.108, X.109, X.110, X.111, X.112, X.113, X.114, X.115, XI.17, XI.23, XI.27, XI.31, XI.32, XI.33, XI.34, XI.36, XI.37, XI.39, XII.1, XII.2, XII.3, XII.4, XII.5, XII.6, XII.7, XII.8, XII.9, XII.10, XII.11, XII.12, XII.13, XII.14, XII.15, XII.17, XII.18, XIII.1, XIII.4, XIII.5, XIII.6, XIII.8, XIII.9, XIII.10, XIII.11, XIII.13, XIII.16, XIII.17, XIII.18

See Commentary on Proposition V.9.