1. ## A square and triangles problem: help, please

Given: square ABCD. E is a point on BC; F is a point on CD.
A straight line is drawn from A to E; a straight line is drawn from E to F; a straight line is drawn from A to F.

Hence, four triangles are now produced: ABE, ECF, ADF and AEF.
The area of ABE is known: A1;
The area of ECF is known: A2;
The area of ADF is known: A3.

The problem is to find the area of triangle AEF.

There are many ways that the solution this problem can be approached.
My chosen method was to determine the lengths of AE, EF and AF and thus calculate the area of triangle AEF, using the well-known general formula:
A = √s(s-p)(s-q)(s-r), where the sides of the triangle are p, q and r and s = semi-sum of p, q and r.

To facilitate the necessary algebra, I allocated lengths as follows:
DF = a; CF = b; BE = c.
Hence, CE = a + b -c.

So we have three unknowns, a, b and c and three knowns: A1, A2 and A3.
Thus the three unknowns are determinable. Hence the area of triangle AEF can be found - as is required.

I started by using the 'area of a triangle = half base x height' for each of the three triangles whose areas are known. This produced the three following equations:
2*A1 = (a+b)*c
2*A2 = (a+b-c)*b;
2*A3 = (a+b)*a.
where * signifies multiply.

All of which looks like a promising start. Unfortunately, having performed the usual elimination of two unknowns, thus enabling the third unknown to be determined, I arrived at a hideous cubic equation for that last unknown.
At that point, for me, everything stopped - and I came here!

So: is there a better route to choose in order to attempt to solve this problem?
And if there is, can you show me, please?

Thank you.

Al. (Skywave) / June 16, 2017

2. ## Re: A square and triangles problem: help, please

I feel like there are missing details in the question.

Does the question ask for an area expression only using certain terms? ie. 'Express the area of triangle AEF only using..."
When the question states that E is a point on BC, does it mean E is on the segment defined by BC, or on the extended line BC?

The area of triangle AEF when E and F are on the line segments BC and CD respectively is Area(Square ABCD) - A1 - A2 - A3

3. ## Re: A square and triangles problem: help, please

Originally Posted by Skywave
Given: square $ABCD$. E is a point on BC; F is a point on CD. A straight line is drawn from A to E; a straight line is drawn from E to F; a straight line is drawn from A to F.
Hence, four triangles are now produced: $ABE, ~ECF, ~ADF ~\&~ AEF$.
The area of $\Delta ABE$ is known: $A_1$;
The area of ECF is known: $A_2$;
The area of ADF is known: $A_3$.
The problem is to find the area ,$\mathscr{A}$ of triangle $\Delta AEF$.
For the sake of notation:
$s$ is the length of the side of square $ABCD$.
$x$ is the length of $\overline{BE}$
$y$ is the length of $\overline{DF}$
It is easy to show that the area of $\Delta AEF$ is: $\Large\mathscr{A}=\dfrac{s^2-xy}{2}$

Now I cannot say from reading this question how many of those variable values are known.