# Area of Triangle in Terms of Exradii

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## Contents

## Theorem

The area of a $\triangle ABC$ is given by the formula:

- $(ABC) = \rho_a \left({s - a}\right) = \rho_b \left({s - b}\right) = \rho_c \left({s - c}\right) = \rho s = \sqrt {\rho_a \rho_b \rho_c \rho}$

where:

- $s$ is the semiperimeter

- $I$ is the incenter

- $\rho$ is the inradius

- $I_a, I_b, I_c$ are the excenters

- $\rho_a, \rho_b, \rho_c$ are the exradii from $I_a, I_b, I_c$, respectively.

## Proof

### Proof of the First Part

First, we show that the area is equal to $\rho_a \left({s - a}\right) = \rho_b \left({s - b}\right) = \rho_c \left({s - c}\right)$.

We pick an excircle, WLOG $I_a$.

\(\displaystyle (ABC)\) | \(=\) | \(\displaystyle (ABI_a) + (ACI_a) - (CBI_a)\) | (see figure above) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {c \rho_a} 2 + \frac {b \rho_a} 2 - \frac {a \rho_a} 2\) | Area of Triangle in Terms of Side and Altitude | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \rho_a \frac {b + c + a} 2 - \rho_a a\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \rho_a s - \rho_a a\) | |||||||||||

\(\displaystyle (ABC)\) | \(=\) | \(\displaystyle \rho_a \left({s - a}\right)\) |

A similar argument can be used to show that the statement holds for the other excircles.

$\Box$

### Proof of the Second Part

Second, we show the area is equal to $(ABC) = \rho s$.

We take the incircle with incenter at $I$ and inradius $\rho$:

\(\displaystyle (ABC)\) | \(=\) | \(\displaystyle (ABI) + (BCI) + (CAI)\) | (see figure above) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {a \rho} 2 + \frac {b \rho} 2 + \frac {c \rho} 2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\rho} 2 \left({a + b + c}\right)\) | |||||||||||

\(\displaystyle (ABC)\) | \(=\) | \(\displaystyle \rho s\) |

$\Box$

### Proof of the Third Part

Finally, we show that the area is equal to $\sqrt{\rho_a \rho_b \rho_c \rho}$:

\(\displaystyle (ABC)^4\) | \(=\) | \(\displaystyle \rho_a \left({s - a}\right) \rho_b \left({s - b}\right) \rho_c \left({s - c}\right) \rho s\) | |||||||||||

\(\displaystyle (ABC)^4\) | \(=\) | \(\displaystyle s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \rho_a \rho_b \rho_c \rho\) | |||||||||||

\(\displaystyle (ABC)^4\) | \(=\) | \(\displaystyle (ABC)^2 \rho_a \rho_b \rho_c \rho\) | Heron's Formula | ||||||||||

\(\displaystyle (ABC)^2\) | \(=\) | \(\displaystyle \rho_a \rho_b \rho_c \rho\) | |||||||||||

\(\displaystyle (ABC)\) | \(=\) | \(\displaystyle \sqrt{\rho_a \rho_b \rho_c \rho}\) |

$\blacksquare$

## Also see

- Heron's Formula, to which this is closely related.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*: Chapter $\text {B}.1$: Appendix