Next, let BD be to DC as BA is to AC; and join AD: the exterior angle CAE shall be bisected by the straight line AD.
For, let the same construction be made.
Then BD is to DC as BA is to AC; [Hypothesis.
and BD is to DC as BA is to AF; [VI. 2.
therefore BA is to DC as BA is to AF; [V. 11.
therefore AC is equal to AF [V. 9.
and therefore the angle ACF is equal to the angle AFC. [1. 5.
But the angle AFC is equal to the exterior angle DAE; [1. 29.
and the angle ACF is equal to the alternate angle CAD; [1. 29.
therefore the angle CAD is equal to the angle DAE; [Ax. 1.
that is, the angle CAE is bisected by the straight line AD.
Wherefore, if the exterior angle &c. q.e.d.
PROPOSITION 4. THEOREM.
The sides about the equal angles of triangles which are equiangular to one another are proportionals; and those which are opposite to the equal angles are homologous sides that is, are the antecedents or the consequents of the ratios.
Let the triangle ABC be equiangular to the triangle DCE, having the angle ABC equal to the angle DCE, and the angle ACB equal to the angle DEC, and consequently the angle BAC equal to the angle CDE the sides about the equal angles of the triangles ABC, DCE, shall be proportionals; and those shall be the homologous sides, which are opposite to the equal angles.
Let the triangle DCE be placed so that its side CE may be contiguous to BC, and in the same straight line with it. [I. 22.
