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S		 Book VI   Proposition 19
But
    as BC is to BG;
    so is the triangle ABC to the triangle ABG; [VI.1]
therefore
    the triangle ABC also has to the triangle ABG a ratio duplicate of that which BC has to EF.

But
    the triangle ABG is equal to the triangle DEF;
therefore
    the triangle ABC also has to the triangle DEF a ratio duplicate of that which BC has to EF.
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