Kitap 1
5
But FC was also proved equal to GB; therefore the two sides BF, FC are equal to the two sides CG, GB respectively; and the angle BFC is equal to the angle CGB, while the base BC is common to them; therefore the triangle BFC is also equal to the triangle CGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CBG.
5
Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG is equal to the angle BCF, the remaining angle ABC is equal to the remaining angle ACB; and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB; and they are under the base.
5
Therefore etc.
5
Q. E. D.
Proposition 6.
6
Enunciation 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.
6
Proof. Let ABC be a triangle having the angle ABC equal to the angle ACB;
6
I say that the side AB is also equal to the side AC.
6
For, if AB is unequal to AC, one of them is greater.
6
Let AB be greater; and from AB the greater let DB be cut off equal to AC the less;
6
let DC be joined.
