Let ABC be an equilateral triangle and AD perpendicular to BC, then AB² + BC² + CA² = ?

AB² = AD² + BD² . . . . . . . (i)
AC² = AD² + CD² . . . . . . . (ii)
AB² + AC² = 2AD² + BD² + CD²
AB² + AC² + BC² = 2AD² + a² + $$\frac{{{{\text{a}}^2}}}{4}$$ + $$\frac{{{{\text{a}}^2}}}{4}$$
AB² + AC² + BC² = 4AD²
          $$\left( {{{\text{a}}^2} - \frac{{{{\text{a}}^2}}}{4} = {\text{A}}{{\text{D}}^2}} \right)$$