Areas

Is there any relationship between the area of Morley’s triangle and the area of the initial triangle? Try the following applet:

(Click on any of the vertices of the triangle \([ABC]\) and move it)

We can immediately conclude that the areas are not directly proportional. In fact, the area of the initial triangle cannot be determined by the area of the Morley’s triangle, since there are an infinite number of triangles that give rise to the same equilateral triangle and with quite different values for their areas (see "the other way around"). However, it is possible to prove that the maximum value of the ratio between the areas is \(\frac{64}{3} \sin^{6} \frac{\pi}{9}\) (approximately \(0.03415\) or \(3.415\%\)), and this value is obtained only when the initial triangle is also equilateral.

Using the applet again, see if you can reach the maximum ratio between the areas (hint: firstly, make one side of the triangle horizontal and then look for the position of the opposite vertex so that the triangle is equilateral).

Note: it is also possible to prove that the maximum ratio between the perimeters of the triangles is \(\frac{8}{\sqrt{3}} \sin^{3} \frac{\pi}{9}\) (approximately \(0.18479\) or \(18.479\%\)). Where do these values come from?