**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* = *r**R*(sin ∠*A* + sin ∠*B* + sin ∠*C*).

(Below is a dynamic geometry sketch illustrating the result).

Triangle Area Formula in terms of angles, *r* & *R*

**Challenge**

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?

**Related Links**

Some other bicentric constructions

Pitot's Theorem

Tangential Quadrilateral Converse

Concurrent Angle Bisectors of a Quadrilateral

Theorem of Gusić & Mladinić

Cosine-Sine Angle Rule

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Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412

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Created 18 July 2023 by Michael de Villiers, using *WebSketchpad*.