A triangle is being divided by two lines. The newly created smaller triangles have the area shown in the picture. What is the area of the whole triangle?
My progress so far: The two triangles $EJB$ and $BJC$ share the line $BJ$. Additionally, $EJ$ and $JC$ share the same height. Given this and the fact that the area of $EJB$ and $BJC$ are the same, $EJ$ and $JC$ must be of the same length. Therefore, $EJB$ and $BJC$ must be congruent. Am I right so far, and if so, where do I go from here?
You are correct that $EJ$ and $JC$ have the same length. However, this doesn't imply that $EJB$ and $BJC$ are congruent triangles.
Hint: For the next step, connect points $E$ and $K$ with segment $EK$, and deduce that triangles $EJK$ and $KJC$ have the same area. This leaves one triangle whose area you need to determine. Say the area of this remaining triangle is $y$. Look at the two sets of triangles whose bases are $AE$ and $EB$. Now explain why the following equation is true: $$ \frac y{1+3}=\frac{y+1 +1}{3 + 3} $$