Question

in equilateral triangle ABC if AN perpendicular to BC prove that AN square = 3BN square

Answer 1

for this triangle,Pythagorean theorem states that
(AN)^2+(BN)^2=(AB)^2
subtract (BN)^2 from both sides
(AN)^2=(AB)^2-(BN)^2
since ∆ABC is equilateral,
perpendicular segment AN to BC
bisects line segment BC,
therefore,AB=2BN
(AN)^2=(2BN)-(BN)^2
(AN)^2=4(BN)^2-(BN)^2
(AN)^2=3(BN)^2