If E is a point on side CA of an equilateral triangle ABC such that, then AB2 + BC + CA2 =
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