The Pythagorean Theorem - Educational Outreach

Hero was the founder of the Higher Technical School at thethen world-renown center of learning at Alexandria, Egypt.He invented the first steam engine, schematically depictedin Figure 3.8, a device that rotated due to the thrustgenerated by steam exiting through two nozzlesdiametrically placed on opposite sides of the axis ofrotation. Working models of Hero’s steam engine are stillsold today as tabletop scientific collectables.In his role as an accomplished mathematician,Heron (as he is better known in mathematical circles)discovered one of the great results of classical mathematics,a general formula for triangular area known as Heron’sTheorem. Heron did not originally use the PythagoreanTheorem to prove his result. Both the Pythagorean Theoremand Heron’s Theorem are independent and co-equal resultsin that each can be derived without the use of the other andeach can be used to prove the other. In this section, we willshow how the Pythagorean Theorem, coupled with ‘heavy’analytic geometry, is used to render the truth of Heron’sTheorem. First, we state Heron’s historic result.Heron’s Theorem: Suppose a general triangle (a trianglehaving no special angles such as right angles) has threesides whose lengths are a , b , and c respectively. Lets [ a b c]/ 2 be the semi-perimeter for the same. Thenthe internal area A enclosed by the general triangle, Figure3.9, is given by the formula:A s( s a)(s b)(s c).ac xh xcbFigure 3.9: Diagram for Heron’s Theorem105