## Three triangles puzzle

May 1, 2020

I stumbled upon this delightful little geometry puzzle on the DataGenetics blog:

There are three identical triangles with bases aligned. If each
triangle has an area *A*, what
is the total area of the two regions (*B*+*C*) shaded in green?

### My solution

My solution relies heavily on finding similar shapes and their scale factors. For clarity, I'm glossing over proving that the shapes I claim are similar actually are, but those details could easily be filled in for rigor.

First, we can establish a relation between the areas *B* and *C*: In the figure below, where we
have added an auxiliary line on the top of the row of the original
triangles (or two more similar upside-down triangles, depending on how
you choose to look at it), the red and blue shapes are similar. The blue
triangle has the upped edge twice as long as the red one, so the scale
factor is two. Therefore the area *B* = 4*C*.

In the next figure, the red triangle has the same altitude as *A* but the base is three times as
long; Therefore its area is 3*A*. The ratio of the red triangle to
the big gray triangle is the same as the ratio *D* : *A*, again because of
similarity of the shapes. The edges of the big gray triangle are three
times those of *A*, so it has
the area 9*A*. Hence *D* : *A* = 3*A* : 9*A* = 1 : 3.
As *B* + *D* = *A*, that
leaves *B* = 2/3 ⋅ *A*.

Now that we know that *C* = 1/4 ⋅ *B* and *B* = 2/3 ⋅ *A*, the rest is
simple arithmetic: $$
B + C = \bigg(1 + \frac{1}{4}\bigg) B = \frac{5}{4} B = \frac{5}{4}
\cdot \frac{2}{3}A = \frac{5}{6}A.
$$