If E is a point on side CA of an equilateral triangle ABC such that , then AB 2 + BC + CA 2 =

Question

Philoid · Accepted Answer

Given in equilateral ΔABC, BE ⊥ AC.

We know that in an equilateral triangle, the perpendicular from the vertex bisects the base.

∴ CE = AE = AC/2

In ΔABE,

⇒ AB² = BE² + AE²

Since AB = BC = AC,

⇒ AB² = BC² = AC² = BE² + AE²

⇒ AB² + BC² + AC² = 3BE² + 3AE²

Since BE is an altitude,

BE = √3 AE

⇒ AB² + BC² + AC²

= 3BE² + BE²

∴ AB² + BC² + AC² = 4BE²