Circumradius and area, R = abc/(4S)

Dependencies: Central angle is twice the inscribed angle on the same arc, Triangle area = ½ × base × height ($S=\tfrac12\cdot$base$\cdot$height), Definition of sin / cos / tan (definition of sine in a right triangle).

Statement

Let $\triangle ABC$ have sides $BC=a$, $CA=b$, $AB=c$, area $S$, and circumradius $R$. Then

The equivalent form $abc = 4RS$ welds together the "product of the three sides" and "area × circumradius", and is the bridge to later results like Euler's formula $OI^2=R^2-2Rr$ and the law of sines $\dfrac{a}{\sin A}=2R$.