If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.
Let ABC be a triangle
having the angle ABC equal to
the angle ACB;
I say that the side AB is
also equal to the side AC.
For, if AB is unequal to AC, one of them is greater.
Let AB be greater; and
from AB the greater
let DB
be cut off equal to AC
the less;
let DC be joined.
Then, since DB is equal
to AC, and
BC is common,
the two sides DB,
BC are equal to the
two sides AC,
CB respectively;
and the angle DBC is equal
to the angle ACB;
therefore the base DC is equal
to the base AB,
and the triangle DBC will be equal
to the triangle ACB,
the less to the greater;
which is absurd.
Therefore AB is not unequal
to AC;
it is therefore equal to it.
Therefore etc. Q.E.D.
Book I: Euclid, Elements, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
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