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/
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