PROPOSITION 3.
If an angle of a triangle be bisected and the straight line cutting the angle cut the base also,
the segments of the base will have the same ratio as the remaining sides of the triangle; and,
if the segments of the base have the same ratio as the remaining sides of the triangle,
the straight line joined from the vertex to the point of section will bisect the angle of the triangle.
Let
ABC be a triangle, and let the angle
BAC be bisected by the straight line
AD; I say that, as
BD is to
CD, so is
BA to
AC.
For let
CE be drawn through
C parallel to
DA, and let
BA
be carried through and meet it at
E.
Then, since the straight line
AC falls upon the parallels
AD,
EC,
the angle ACE is equal to the angle CAD. [I. 29]
But the angle
CAD is by hypothesis equal to the angle
BAD; therefore the angle
BAD is also equal to the angle
ACE.
Again, since the straight line
BAE falls upon the parallels
AD,
EC,
the exterior angle BAD is equal to the interior angle AEC. [I. 29]
But the angle
ACE was also proved equal to the angle
BAD;
therefore the angle ACE is also equal to the angle AEC, so that the side AE is also equal to the side AC. [I. 6]
And, since
AD has been drawn parallel to
EC, one of the sides of the triangle
BCE, therefore, proportionally, as
BD is to
DC, so is
BA to
AE.
But
AE is equal to
AC; [
VI. 2] therefore, as
BD is to
DC, so is
BA to
AC.
Again, let
BA be to
AC as
BD to
DC, and let
AD be joined; I say that the angle
BAC has been bisected by the straight line
A.D.
For, with the same construction, since, as
BD is to
DC, so is
BA to
AC, and also, as
BD is to
DC, so is
BA to
AE : for
AD has been drawn parallel to
EC, one of the sides of the triangle
BCE: [
VI. 2] therefore also, as
BA is to
AC, so is
BA to
AE. [
V. 11]
Therefore
AC is equal to
AE, [
V. 9] so that the angle
AEC is also equal to the angle
ACE. [
I. 5]
But the angle
AEC is equal to the exterior angle
BAD, [
I. 29] and the angle
ACE is equal to the alternate angle
CAD; [
id.]
therefore the angle BAD is also equal to the angle CAD.
Therefore the angle
BAC has been bisected by the straight line
AD.
Therefore etc. Q. E. D.
