Proving Concurrence Using Vectors

Date: 10/17/2005 at 12:55:13
From: Hamsika
Subject: show triangle angle bisectors meet at a point

How do you prove that angle bisectors are concurrent using vectors?  I
have proved this using coordinate geometry, but I do not know how to
find the point of intersection using vectors.

Date: 10/18/2005 at 20:25:41
From: Doctor George
Subject: Re: show triangle angle bisectors meet at a point

Hi Hamsika,

Thanks for writing to Doctor Math.

I have not been able to find an elegant vector proof for you, and the
tedious proof is a bit much to put into a text format.  Here are some
hints on how to proceed.

If A and B are vectors with concurrent tails, then the direction
vector for the bisector line is

    A       B
   ---  +  ---  
   |A|     |B|

To see why, take the dot product of this vector with both A and B.

Knowing this, you can construct parametric vector equations for the
bisector lines.  If you solve for the intersection twice, using two
different pairs of bisectors, you can then show that the two
intersection points are concurrent.  The parameter for the line that
is used in both bisector pairs must be the same in each case.

Here is a hint that will help with the computations.  If U and V are
vectors such that |U| = |V| then (U-V).(U+V) = 0.  You can use this
fact to eliminate a parameter from the equations.

See if you can take it from there.  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/