Proof
Decompose into similar triangles by joining one vertex to all others;
apply VI.19 to each triangle and sum via V.12.
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Depends on (3)
- VI.18Proposition VI.18On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.
- VI.19Proposition VI.19Similar triangles are to one another in the duplicate ratio of the corresponding sides.
- V.12Proposition V.12If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the…
Required by (dependents) (6)
- VI.22Proposition VI.22If four straight lines be proportional, the rectilineal figures similar and similarly described upon them will also be…
- VI.25Proposition VI.25To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal…
- VI.27Proposition VI.27Of all parallelograms applied to the same straight line and deficient by parallelogrammic figures similar and similarly…
- VI.31Proposition VI.31In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly…
- XI.33Proposition XI.33Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides.
- XII.1Proposition XII.1Similar polygons inscribed in circles are to one another as the squares on the diameters.
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