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Book IV   Proposition 13
Therefore
    the angle CBF is equal to the angle CDF.

And, since
    the angle CDE is double of the angle CDF,
and
    the angle CDE is equal to the angle ABC,
while
    the angle CDF is equal to the angle CBF;
therefore
    the angle CBA is is also double of the angle CBF.

Therefore
    the angle ABF is equal to the angle FBC;
therefore
    the angle ABC has been bisected by the straight line BF.
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