What is the circumradius of an equilateral triangle of sides 8 centimetres?

Question

Philoid · Accepted Answer

Let the equilateral triangle be Δ ABC, Circumcentre and circumradius be E and r respectively.

We know that in equilateral triangle median and internal angle bisector are same.
Here AE, BE and CE are part of medians.
⇒ EA acts as an internal angle bisector for ∠CAB.
⇒ ∠CAE = ∠EAB

In Δ ABC,
∠CAB = ∠CAE + ∠EAB = 60°
⇒ ∠CAB = 2(∠CAE ) = 2(∠EAB) = 60°
⇒ ∠CAE = ∠EAB = 30°

Extend CE to meet at AB at H.
CH is a median as well as altitude.
⇒ CH ⊥ AB and ∠CHA = ∠CHB = 90°

In Δ AEH,

∠EAH + ∠AHE + ∠HEA = 180°
⇒ 30° + 90° + ∠HEA = 180°
⇒ ∠HEA = 60°

We know that sides of any triangle of angles 30°, 60° and 90°
are in the ratio 1: √3: 2 .

⇒ EH :AH :AE = 1: √3: 2
⇒ EH :4 :r = 1: √3: 2
⇒ 4 :r = √3: 2