# Smartest Triangles

#### anemone

##### MHB POTW Director
Staff member
Let the IQ of a triangle be the ratio $$\displaystyle \frac{\text{area of the triangle}}{(\text{perimeter of the triangle})^2}$$.

This is a dimensionless number. Show that the smartest triangles are equilateral triangles.

#### MarkFL

Staff member
Using Heron's formula, one notices by symmetry that interchanging any two of the variables representing the side lengths results in the same area and perimeter functions, hence Lagrange multipliers used with:

The objective function $$\displaystyle A(x,y,z)=\frac{\sqrt{s(s-x)(s-y)(s-z)}}{4s^2}$$ where $$\displaystyle s=\frac{x+y+z}{2}$$

Subject to the constraint $$\displaystyle g(x,y,z)=x+y+z-p=0$$

will necessarily lead to the implication:

$$\displaystyle x=y=z=\frac{p}{3}$$

Thus, the triangle having the greatest IQ is equilateral.

#### anemone

##### MHB POTW Director
Staff member
Hey MarkFL, this is the fastest solution that I have ever received so far!

Great job!

#### MarkFL

Staff member
Let's just say this problem is nearly identical to a recent university level POTW...copy/paste/edit/done.

#### anemone

##### MHB POTW Director
Staff member
Looks to me these copy/paste/edit things are signs of you're on the verge of cheating...