The Pythagorean Theorem - Educational Outreach

Some historians believe Stewart’s Theorem can be traced asfar back as Archimedes. Other historians believe thatSimpson (one of Stewart’s academic mentors) actuallyproved ‘Stewart’s Theorem’. Controversy aside, Stewart wasa brilliant geometer in his own right. Like Euclid, Stewartwas an expert organizer of known and useful results in theburgeoning new area of mathematical physics—especiallyorbital mechanics.Unlike the three previous theorems, Stewart’sTheorem is not co-equal to the Pythagorean Theorem. Thismeans that Stewart’s Theorem requires the use of thePythagorean Theorem in order to construct a proof.Independent proofs of Stewart’s Theorem are not possibleand, thus, any attempt to prove the Pythagorean Theoremusing Stewart’s Theorem becomes an exercise in circularlogic. We now proceed with the theorem statementmad hx nc m nbFigure 3.14: Diagram for Stewart’s TheoremStewart’s Theorem: Suppose a general triangle (atriangle having no special angles such as right angles) hassides with lengths a , b , and c . Let a line segment ofunknown length d be drawn from the vertex of the triangleto the horizontal side as shown in Figure 3.14, cutting cinto two portions m and n where c m n . Then, thefollowing relationship holds among the six line segmentswhose lengths arena2a , b,c,d,m , and n :2 2 mb c[d mn].113