Area of a Triangle in Terms of Exradii

Theorem

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

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

In this formula:

• $s$ 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 $I_a$.

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)\)	\(=\)	\(\displaystyle (ABI_a) + (ACI_a) - (CBI_a)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	(see figure above)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {c \rho_a} 2 + \frac {b \rho_a} 2 - \frac {a \rho_a} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by Area of a Triangle in Terms of Side and Altitude
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \rho_a \frac {b + c + a} 2 - \rho_a a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \rho_a s - \rho_a a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)\)	\(=\)	\(\displaystyle \rho_a \left({s - a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

A similar argument can be used to show that the statement holds for the others 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 $\rho$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)\)	\(=\)	\(\displaystyle (ABI) + (BCI) + (CAI)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	(see figure above)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {a \rho} 2 + \frac {b \rho} 2 + \frac {c \rho} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\rho} 2 \left({a + b + c}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)\)	\(=\)	\(\displaystyle \rho s\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\Box$

Proof of the Third Part

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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)^4\)	\(=\)	\(\displaystyle \rho_a \left({s - a}\right) \rho_b \left({s - b}\right) \rho_c \left({s - c}\right) \rho s\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)^4\)	\(=\)	\(\displaystyle s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \rho_a \rho_b \rho_c \rho\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)^4\)	\(=\)	\(\displaystyle (ABC)^2 \rho_a \rho_b \rho_c \rho\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	(by Heron's Formula)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)^2\)	\(=\)	\(\displaystyle \rho_a \rho_b \rho_c \rho\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (ABC)\)	\(=\)	\(\displaystyle \sqrt{\rho_a \rho_b \rho_c \rho}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Comment

This formula for the area of a triangle is closely related to Heron's Formula.

Sources

• George F. Simmons: Calculus Gems (1992), Chapter $\text {B}.1$: Appendix