Triangle Area Formula
Here's a little known formula for the area of a triangle ABC in terms of its angles, the radius r of its incircle and the radius R of its circumcircle: area ∆ABC = rR(sin ∠A + sin ∠B + sin ∠C).
(Below is a dynamic geometry sketch illustrating the result).
Triangle Area Formula in terms of angles, r & R
Can you explain why (prove that) this area formula is true?
Further Generalization Bicentric Polygons
1) Can you generalize further to a bicentric quadrilateral (a quadrilateral with an incircle as well as a circumcircle)?
(Click on the 'Link to Bicentric quadrilateral' button on the bottom right to check your conjecture.)
2) Can you generalize further to any bicentric polygon? Can you provide a general proof for any bicentric polygon?
Created 18 July 2023 by Michael de Villiers, using WebSketchpad.