The Angle Bisector and Equal Side Ratios

Date: 05/17/98 at 17:17:07
From: Jason Yerep
Subject: Geometry Proof

Please help me to solve this proof.

Given: Triangle ABC
       Angle bisector BD

           B
          /|\
     x   / | \  y
        /  |  \
       /   |   \
      /    |    \
     /     |     \
   A/      |      \C
    ---------------
      a    D   b

Prove:

    x     y
   --- = ---
    a     b  

where x = side AB, a = segment AD, y = side BC, and b = segment DC.

Date: 05/19/98 at 10:06:06
From: Doctor Santu
Subject: Re: Geometry Proof

Dear Jason:

This problem can be solved by comparing the areas of the two 
triangles. The crucial idea is that angle bisectors are equally 
distant from both sides of the angle. In other words, if you have an 
angle, and you draw the angle bisector, no matter which point on the 
bisector you pick, it will be the same distance from the two lines 
of the original angle. This is the main idea in the solution I came 
up with.

Below is a figure drawn so that it doesn't look like an isosceles 
triangle. 

    

See the blue lines? They're perpendiculars from the point D to the two 
sides of the top angle. Since D is on the angle bisector, it's the 
same (perpendicular) distance from the two sides BA and BC. How do we 
know this? The two triangles with BD as a common side and containing 
the blue lines are congruent (SAA) so the blue lines, being 
corresponding parts, must be congruent.

Now forget the blue lines for the moment -- imagine that they're not 
there. Ignore the red line, too, and what you have left is the 
original triangle you drew, with the line down the middle. Let's call 
the areas of the two triangles A1 and A2. Then:

   A1 = (1/2)*base*height = (1/2)*a*h

where h is the altitude of the left triangle, the length of the red 
line. Also:

   A2 = (1/2)*b*h 

since the triangles have the same altitude, from the way their bases 
are collinear, and have a vertex in common.

But you can figure the two areas in different ways, too. The triangle 
on the left can be turned around so that the blue line on the left 
is an altitude. Now its area is:

   A1 = (1/2)*x*k

where k is the length of the blue line.

In the same way, the area of the triangle on the right is:

   A2 = (1/2)*y*k

So a*h = x*k and b*h = y*k. A simple calculation shows that:

   x/y = a/b

Good luck,

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