Pons Asinorum

The fifth proposition in Book I of Euclid’s Elements states that the two base angles of an isosceles triangle are equal (in the figure below, angles B and C).

For centuries, this result has been known as Pons Asinorum, or the Bridge of Asses, apparently a metaphor for a problem that separates bright sparks from dunces. Euclid proved the proposition by extending the sides AB and AC and drawing lines to form additional triangles. His proof is quite complicated.

Pons Asinorum. Figure from Oliver Byrne's Elements of Euclid.

Pons Asinorum. From Oliver Byrne’s Elements of Euclid.

A simpler approach, popular for a hundred years or so, is to draw the line that bisects the apex angle A, splitting the triangle into two parts, which are then shown to be congruent, or equal in all respects. This requires use of an earlier result, Euclid’s proposition I.4, which says that two triangles are congruent if they have two sides and the included angle equal.

An Ingenious Proof

Around 1960 another proof appeared, allegedly discovered by a computer. It ingeniously compared the triangle ABC to its mirror image ACB (see figure below) and used Proposition I.4 to show that they are congruent.

Isosceles triangle ABC and its mirror image.

Isosceles triangle ABC and its mirror image.

This “new” proof is intriguing in that it treats the triangle and its mirror image as separate for purposes of deduction but identical for purposes of conclusion:

AB (in ABC) = AC (in ACB)

AC (in ABC) = AB (in ACB)

A (in ABC) = A (in ACB).

Therefore ABC and ACB are congruent

Thus, angle B equals angle C.         QED.

When this proof appeared around 1960, it was considered by many people as demonstrating that computers can be creative. The proof was frequently cited as evidence of artificial intelligence (AI), for example in Douglas Hofstadter’s remarkable book Gödel, Escher, Bach: an Eternal Golden Braid.

But Michale Deakin of Monash University has investigated the matter. He reports an interview in 1981 in the New Yorker, in which AI guru Marvin Minsky of MIT stated that he produced the proof himself “by hand simulation of what a machine might do”.

Amazingly, the ingenious proof was first discovered by the last great Greek geometer, Pappus of Alexandria, working around 320 AD. It was derided by the nineteenth century Oxford mathematician C. L. Dodgson, who’s imagined reaction of Euclid was: “Surely that has too much of the Irish Bull about it”. Dodgson was none other than Lewis Carroll, author of Alice in Wonderland.

The Standard Proof

But what of the “standard proof” using the bisector of the apex angle? Deakins points out that the reasoning in A School Geometry, a book by Hall and Stevens that some of us slaved over long ago, is circular.

The proof of Pappus, rediscovered by Minsky and wrongly attributed to a computer, is certainly elegant. But perhaps it is safest to stick with Euclid’s original proof. At least one school child produced the Pappus proof in an examination and was marked wrong for it.


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