Triangle Area Proofs

Date: 01/23/2002 at 01:14:56
From: N P JOSHI
Subject: Problems of relations of area of Triangles

Triangle ABC is equilateral, with D at the midpoint of BC. 
Construct BE parallel to AC. Triangle BDE is also equilateral.
Join AE to BC at F.  The area of triangle BDE is equal to 1/4 the area 
of triangle ABC and 1/2 the area of triangle BAE.

Prove that:
	
 (i) area of triangle BFE = 2(area of triangle FED)
(ii) area of triangle FED = 1/8th(area of triangle AFC)

This is a problem for a standard IX students who are yet to be taught 
about analytical geometry or concepts of coordinates. The question 
needs to be solved with the help of parallel lines or otherwise. 

Date: 01/23/2002 at 16:51:37
From: Doctor Jaffee
Subject: Re: Problems of relations of area of Triangles

Hi NP -

Consider triangle ABF and triangle EDF. Can you see that they are 
similar with the ratio of corresponding sides equal to 2:1? If you can 
prove that, then you should be able to prove that the area of triangle 
BFE is twice the area of triangle FED. Once you've accomplished that, 
then the last part should be fairly easy.

But I would take an analytic approach to this problem, rather than a 
synthetic approach. In other words, this problem can be solved using a 
coordinate system rather than using the theorems about parallel lines, 
equilateral triangles, etc.

Suppose you were to draw triangle ABC so that B is at point (0,0), C 
is at (2,0), and A is at (1,sqrt(3)). Then you have an equilateral 
triangle each of whose sides has length 2. If you want to generalize 
the problem, you could let C be at point (c,0). Then A would have the 
coordinates (c/2,c*sqrt(3)/2). But I don't think that is necessary.  
When you construct the equilateral triangle BED, each of its sides 
will have length 1 since BD = 1. Therefore, the coordinates of E are 
(1/2,-sqrt(3)/2).

My suggestion, then, is find the equation of the line that goes  
through A and E. Find the equation of the line that goes through B 
and D. Then find the intersection of those two lines, point F. Once 
you have that information, the rest is easy.

Give it a try and if you want to check your solution, write back. If 
you are having difficulties, let me know what you have done so far and 
I'll help you out.

Good luck.

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