<P> and the generalized form follows . </P> <P> The angle bisector theorem appears as Proposition 3 of Book VI in Euclid's Elements . According to Heath (1956), p. 197 (vol. 2), the corresponding statement for an external angle bisector was given by Robert Simson who claimed that Pappus assumed this result without proof . Heath goes on to say that Augustus De Morgan proposed that the two statements should be combined as follows: </P> <Dl> <Dd> If an angle of a triangle is bisected internally or externally by a straight line which cuts the opposite side or the opposite side produced, the segments of that side will have the same ratio as the other sides of the triangle; and, if a side of a triangle be divided internally or externally so that its segments have the same ratio as the other sides of the triangle, the straight line drawn from the point of section to the angular point which is opposite to the first mentioned side will bisect the interior or exterior angle at that angular point . </Dd> </Dl> <Dd> If an angle of a triangle is bisected internally or externally by a straight line which cuts the opposite side or the opposite side produced, the segments of that side will have the same ratio as the other sides of the triangle; and, if a side of a triangle be divided internally or externally so that its segments have the same ratio as the other sides of the triangle, the straight line drawn from the point of section to the angular point which is opposite to the first mentioned side will bisect the interior or exterior angle at that angular point . </Dd>

In a triangle abc d is a point on bc such that bd/dc=ab/ac