G.183 equal magnitudes ratio

V.7

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.
equal_magnitudes_ratio(A, B, C)
given A, B be = magnitudes and C any other, chance, magnitude
prove each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A, B
A B C D E F
given A, B be = magnitudes and C any other, chance, magnitude
construct Let equimultiples D, E of A, B be taken, and of C another, chance, multiple F.
assert D is the same multiple of A that E is of B → while A = B,
assert D = E
assert F is another, chance, magnitude
assert If therefore D is in excess of F, E is also in excess of F, if = to it, =
assert if less, less
assert D, E are equimultiples of A, B, while F is another, chance, multiple of C
∵ magnitudes_said_ratio_defassert as A is to C, so is B to C
assert I say next that C also has the same ratio to each of the magnitudes A, B
assert With the same construction, we can prove similarly that D = E
assert F is some other magnitude
assert If therefore F is in excess of D, it is also in excess of E, if =, =
assert if less, less
assert F is a multiple of C, while D, E are other, chance, equimultiples of A, B
∵ magnitudes_said_ratio_def∴ as C is to A, so is C to B

From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely.

Dependency Graph

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Required for

G.188, G.192, G.193, G.195, G.199, G.201, G.202, G.204, G.205, G.206, G.207, G.208, G.210, G.211, G.212, G.213, G.214, G.215, G.216, G.218, G.219, G.220, G.221, G.222, G.223, G.225, G.226, G.227, G.228, G.229, G.231, G.232, G.234, G.235, G.236, G.237, G.238, G.239, G.240, G.241, G.243, G.244, G.245, G.273, G.278, G.279, G.280, G.281, G.285, G.289, G.290, G.292, G.295, G.297, G.298, G.299, G.301, G.303, G.304, G.305, G.306, G.314, G.315, G.316, G.317, G.322, G.323, G.324, G.325, G.326, G.327, G.328, G.329, G.330, G.331, G.332, G.333, G.334, G.335, G.337, G.338, G.339, G.342, G.343, G.344, G.345, G.348, G.349, G.364, G.370, G.376, G.377, G.378, G.379, G.381, G.382, G.383, G.384, G.385, G.386, G.389, G.390, G.393, G.394, G.395, G.396, G.397, G.398, G.399, G.400, G.401, G.402, G.403, G.404, G.405, G.406, G.407, G.408, G.409, G.410, G.411, G.412, G.413, G.414, G.415, G.416, G.417, G.418, G.419, G.420, G.421, G.423, G.425, G.427, G.429, G.431, G.433, G.434, G.435, G.436, G.437, G.438, G.439, G.440, G.441, G.442, G.443, G.444, G.445, G.446, G.447, G.448, G.449, G.450, G.451, G.452, G.453, G.457, G.458, G.459, G.460, G.461, G.462, G.463, G.464, G.465, G.466, G.467, G.468, G.470, G.472, G.474, G.476, G.477, G.478, G.479, G.480, G.481, G.482, G.483, G.484, G.485, G.486, G.487, G.488, G.489, G.490, G.491, G.492, G.493, G.494, G.495, G.496, G.497, G.498, G.499, G.500, G.501, G.520, G.528, G.536, G.541, G.542, G.543, G.544, G.546, G.547, G.550, G.551, G.552, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578, G.579, G.580, G.582, G.583, G.584, G.586, G.587, G.588, G.589, G.592, G.597, G.600, G.601