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December 13, 2003

Small Problem

Following the overwhelming popularity (joke!) of a couple of geometrical puzzles posted by Ole and myself, here's another one.

Ivor Grattan-Guinness's Fontana History of the Mathematical Sciences, p. 566, mentions a neat little theorem in Euclidean geometry.

Draw a square, ABCD. On one of the sides, AB, construct a right-angled triangle AFB, with the hypoteneuse on AB, the vertex at F outside the square, and the angle AFB a right angle. Now bisect the right angle, and extend the bisecting line across the square.

The theorem states that the bisecting line also bisects the square, i.e. divides it into two portions equal in area.

As G-G says, it is a 'lovely' theorem, and remarkably it doesn't seem to have been stated before the 1840s.

But - alas - G-G doesn't give any indication of a proof.

So can you prove it, by elementary Euclidean geometry? (No need to go right back to the axioms, but state any intermediate propositions, e.g. 'angles at base of an isosceles triangle are equal').

I will post my own proof in a few days' time. Given the simplicity of the theorem, I thought it would be easy to prove. But I didn't find it that easy, and my proof is quite complicated. Hopefully someone will find a simpler one.

Posted by David B at 04:18 AM