Angle-bisector Proof

Date: 10/16/97 at 14:46:12
From: Tim Gelfand
Subject: Geometry

Dear Dr. Math,

   Prove that in a triangle ABC, a pair of angle-bisectors cannot be 
   perpendicular.  

   I have absolutely no idea what this is even about. Haven't done 
   geometry in 15 years!

Yours truly,
Tim in Montreal

Date: 10/16/97 at 15:47:35
From: Doctor Wilkinson
Subject: Re: Geometry

Draw a picture! 

Suppose the bisector of the angle at A and the bisector of the angle 
at B are perpendicular. Suppose they intersect at D. This gives you a 
smaller triangle ABD, where the angle at D is a right angle. That 
tells you that the angle DAB and the angle DBA add up to 90 degrees, 
because the angles of a triangle always add up to 180 degrees. That 
means that the angle CAB, which is twice DAB (because AD is the angle 
bisector; and the angle CBA, which is twice DBX (Because BD is the 
angle bisector), add up to twice 90 degrees, or 180 degrees. But 
that's impossible, because all three angles of the triangle have to 
add up to 180 degrees, and there's nothing left.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/