A triangle is split into three parts as shown on the picture...

Question

A triangle is split into three parts as shown in the picture (not drawn to scale). The areas of the three regions form an arithmetic progression (where the area of quadrilateral $A_2$ is the middle term). Using the information in the picture, find the length of the missing side of triangle $A_1$.

Answer 1

$$A_{1}+A_{3}=2A_{2}$$

I used A and B to denote the areas of certain triangles in the figure below (I regret choosing A as a variable, it has nothing to do with the areas given in the problem statement). Let A be the leftmost triangle and let B be the triangle which is a part of $A_{2}$.

Split the areas according to the bases (since the triangles have the same height) and plug the expressions into the above equation:

$$\frac{1}{2} \left(A + \left(\frac{1}{2} (4 A + 3 B) + 2 B\right)\right) = 3 A + B, $$

$$B=2A$$

Also from area division note that:

$$\frac{3B}{3}=\frac{4A}{?}\ $$

$$2=\;?$$