How to find radius of circumscribed circle about an obtuse triangle?

Question

How to find radius of circumscribed circle about an obtuse triangle?

Answer 1

The full form of the Sine Rule is:

a / sin(A) = b / sin(B) = c / sin(C) = 2R

where R is the radius of the circumcircle.

This is true for all triangles, whether obtuse- or acute-angled.

Join vertices B and C of triangle ABC to the centre O of the circle.

From O drop a perpendicular on to BC meeting it at M.

Angle A = (1 / 2) angle BOC (angle at circumference and at centre)

Angle BOM = (1 / 2) angle BOC (triangle BOC isosceles with OB = OC = R)

Therefore:

Angle BOM = angle A

From triangle OMB:

sin(BOM) = BM / OB

sin(A) = BM / R

BM = BC / 2

sin(A) = BC / (2R)

sin(A) = a / 2R

a / sin(A) = 2R.

Similarly b / sin(B) = 2R and c / sin(C) = 2R.