Consider a triangle ABC. Let the angle bisector of angle A intersect side BC at a point D between B and C.
The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC.
BD/DC= AB/AC
Prove it.
VECTOR SOLUTION
Two uppercase letters such as PQ will denote a
vector from point P to point Q. Its length will
be denoted by PQ. Single lowercase letters
will represent real numbers. In our problem we
have a = BC, b = AC, and c = AB.
AD = t*( AB/c + AC/b ) for some t
AD = w*AB + (1w)*AC for some w
Combining these two equations gives
AD = ( b*AB + c*AC )/(b+c)

BD = AD  AB = c*( AC  AB )/(b+c) = c*BC/(b+c)
BD = a*c/(b+c)
DC = AC  AD = b*( AC  AB )/(b+c) = b*BC/(b+c)
BD = a*b/(b+c)
Therefore,
BD/DC = c/b = AB/AC
QED
Edited on November 9, 2015, 7:46 pm

Posted by Bractals
on 20151109 13:25:56 