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 + (1-w)*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 2015-11-09 13:25:56