Kaushik Basu finds a new way to prove something every geometry student already knows
News flash: World Bank economist Kaushik Basu has proved that when it comes to right triangles, a² + b² = c². This merits the briefest of footnotes in the annals of mathematics, because the Greek mathematician Pythagoras proved the same theorem around 500 B.C. Every kid in geometry class learns that the sum of the squares of the lengths of the sides of a right triangle equals the square of the length of the hypotenuse. This is no Fermat's Last Theorem.
It's kind of impressive nonetheless. Basu now belongs to an august tradition of people who have found new ways of proving something we have known for 2,600 years. Amazingly, a U.S. president was one of the provers. James Garfield completed his proof of the Pythagorean theorem in 1876, four years before being elected president. (This Khan Academy video gives the presidential derivation.)
Basu, a Cornell University economist, demonstrated his proof in a paper entitled "A New and Very Long Proof of the Pythagoras Theorem By Way of a Proposition on Isosceles Triangles."
"I treat this as my hobby. I do it for fun," Basu said in an interview. Has he told World Bank President Jim Kim about his achievement? Not yet. "I'll have to send him a note assuring him that this was weekend work," Basu laughs.
In deriving the proof Basu discovered some new things about the properties of isosceles triangles — ones with at least two sides of equal length. Right triangles are ones with one square (90-degree) angle. The length of the proof is special, too, he wrote in his paper:
"How then can one justify presenting a new and longer proof of Pythagoras’ theorem? The only way to answer this is to invoke another Greek, Constantine Cavafy and his classic poem, Ithaca, which describes the long journey to Odysseus’ home island. When you reach the island, the poet warns the reader, you are likely to be disappointed, for it will have little new to offer. But do not be disappointed, Cavafy tells the reader, for Ithaca’s charm is the journey itself."