# Complex-number method to minimize equilateral-triangle area inside right triangle of side lengths $2\sqrt3$, $5$, and $\sqrt{37}$

The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $$2\sqrt3$$, $$5$$, and $$\sqrt{37}$$, as shown, is $$\tfrac{m\sqrt{p}}{n}$$, where $$m$$, $$n$$, and $$p$$ are positive integers, $$m$$ and $$n$$ are relatively prime, and $$p$$ is not divisible by the square of any prime. Find $$m+n+p$$.

I am trying to solve this problem just with complex numbers. I let the vertices of the large triangle to be $$a,b,c$$ for $$b=0,a=2\sqrt{3}i,c=5.$$ I let the vertices of the equilateral triangle to be $$z_1,z_2,z_3$$ for $$z_1$$ along $$AB$$, $$z_2$$ along $$AC$$, and $$z_3$$ along $$BC$$. In the first place, it is well-known that $$z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1.$$

It is futile to consider the equations for $$A,Z_1,B$$ and $$B,Z_3,C$$ to be collinear because we set those as the imaginary and real axes. With the equation for $$A,Z_2,C$$ to be collinear, we have $$\frac{z_2-a}{z_2-c}=\frac{\overline{z_2}-\overline{a}}{\overline{z_2}-\overline{c}}=\frac{\overline{z_2}+a}{\overline{z_2}-c},$$

with that step from $$\overline{a}=-a$$ and $$\overline{c}=c,$$ as these are along the imaginary and real axes respectively. With some manipulation, we get to $$a(z_2+\overline{z_2})+c(z_2-\overline{z_2})-2ac=0$$ $$\Longleftrightarrow a\Re(z_2)+c\Im(z_2)i=10\sqrt{3}$$ $$\Longleftrightarrow 2\sqrt{3}x_2+5y_2=10\sqrt{3},$$ for $$\Re(z)=x,\Im(z)=y.$$

But I'm not quite sure what to do now, because I can't think of any synthetic simplifications.

Rewrite the equilateral condition $$z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$$ as $$z_3^2 -(z_1+z_2)z_3 + z_2^2+z_3^2-z_1z_2=0$$ which is quadratic in $$z_3$$, yielding $$z_3= e^{i\frac\pi3}z_1 + e^{-i\frac\pi3}z_2$$ Substitute $$z_3$$ into the line equation below for the hypotenuse
$$\frac{z+\bar z}{10}+ \frac{z-\bar z}{4\sqrt3 i}=1$$ obtained from the given side lengths, leading to $$\frac7{20}|z_1 |+ \frac{11\sqrt3}{60}|z_2 |=1$$ where $$z_1=\bar z_1$$ and $$z_2= -\bar z_2$$ are used. Then, the area of the equilateral triangle is
$$A= \frac{\sqrt3}{4} |z_2-z_1|^2 = \frac{\sqrt3}{4}( |z_2|^2 +|z_1|^2 )\ge \frac{\sqrt3}{4} \frac{\left(\frac7{20}|z_1 |+ \frac{11\sqrt3}{60}|z_2 |\right)^2 }{\left(\frac7{20}\right)^2 + \left(\frac{11\sqrt3}{60}\right)^2 }= \frac{75\sqrt3}{67}$$ where the C-S inequality is applied.
• Great answer, I realized that instead of using the criterion for an equilateral triangle, we can perform a spiral similarity and arrive at the same expression regarding the point on the hypotenuse: $z_3\mapsto e^{i\pi/3}(z_3-z_1)+z_1=z_2,$ where we take $z_2$ on the hypotenuse and $z_1$ on the imaginary axis.