Triangle area formulae involving the inradius and circumradius

While laid up over the last few weeks following surgery, I wrote an article containing some ideas that try to convey clearly how interesting (and comprehensible) these formulae are. I'm not breaking any new mathematical ground here, of course, but I was interested in how well I could illustrate the ideas with diagrams. You be the judge... (rather than try cram the whole thing into a blog post, I've just outlined the main ideas below. Please read the full article for all the details).

The first diagram shows the "base triangle" ABC, with sides labelled a, b and c in the conventional way.

This diagram shows the inscribed circle or incircle of triangle ABC, with radius r (the inradius of the triangle). This is the largest circle that can be drawn inside the trangle and is tangent to all three sides. The formula makes use of the semiperimeter, s = (a + b + c)/2.

The area of the triangle is simply rs. The article contains a couple of different proofs of this.

This diagram shows the circumscribed circle or circumcircle of triangle ABC, with radius R (the circumradius of the triangle). This is the smallest circle that completely contains the triangle, passing through all three vertices.

 

The area of the triangle is abc/4R. The article contains a proof, but also looks at the link between the circumradius and the sine rule and how this could possibly help to demystify why the rule works.

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