Thursday, March 28, 2013

Heron's formula for the area of a triangle

The angle bisectors of the triangle meet at the center of the inscribed circle of radius r.  If we let \(2\alpha=A\), \(2\beta = B\), and \(2\gamma=C\), we have \(\alpha+\beta+\gamma=\frac{\pi}{2}\). 

Let x be the distance from the vertex at A to points of tangency, y the distance from B, and z the distance from C.  Then then lengths of the triangle sides opposite A, B, and C are respectively \(a=y+z\), \(b=x+z\), and \(c=x+y\).

Thus if we name the semiperimeter sthen \(s=x+y+z\), \(x=s-a\), \(y=s-b\), and \(z=s-c\).

\(\tan \alpha =\frac{r}{x}\), \(\tan \beta =\frac{r}{y}\), and \(\tan \gamma =\frac{r}{z}\).  Because \(\gamma\) and \( (\alpha+\beta )\) are complementary angles, we obtain

\[ \tan\left( \frac{\pi}{2}  - (\alpha+\beta) \right)   = \frac{r}{z} \]
\[ \tan\left( \alpha+\beta \right)   = \frac{z}{r} \]
\[ \frac{ \tan \alpha+\tan\beta}{1-\tan\alpha \tan\beta}   = \frac{z}{r} \]
\[r \left(\tan \alpha+\tan\beta  \right)= z (1-\tan\alpha\tan\beta) \]
\[r \left( \frac{r}{x}+\frac{r}{y} \right) = z \left( 1 - \frac{r}{x}\frac{r}{y} \right) \]
\[ r^2 y + r^2 z = xyz - r^2 z \]
\[ r^2 ( x+y+z) = xyz \]
\[ r^2 s = xyz \]

The radii at the points of tangency and the angle bisectors form 3 pairs of congruent triangles.  The area of \(\Delta ABC\) is \(xr+yr+zr= r(x+y+z)\), so area \(=rs\), and \( (\text{area})^2=r^2s^2\).  Using results we have above, we obtain
\[ (\text{area})^2 = s\cdot xyz = s(s-a)(s-b)(s-c)\]
so the area is \(\sqrt{s(s-a)(s-b)(s-c)}\).