October 14

The Three Circle Area Theorems
for Triangles

Circumscribed Circle Area Theorem | Inscribed Circle Area Theorem
Escribed Circle Area Theorem | Inscribed and Escribed Area Corollaries
Heron's Theorem | Brahmagupta's Theorem

Circumscribed Circle Area Theorem

Let be any triangle and denote the lengths of the sides
by AB = c, BC = a, and AC = b.

let K = the area of

Then

K = abc/4R

where R = the circumradius.of the circumscribed circle.

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Inscribed Circle Area Theorem

Let be any triangle,

let K = the area of ,
let r = the inradius, and
s = the semiperimeter.

Then:

K = sr

11

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Escribed Circle Area Theorem

Let be any triangle,

let K = the area of ,
let ra = the exradius of the escribed circle opposite <A,
let BC = a, and
let s = the semiperimeter.

Then:

K = (s - a) ra

ttt

UEST

Inradius and Exradius Area Corollaries

Corollary 1

If r is the inradius of and K is its area, then
r * ra * rb *rc = K2

Corollary 2

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