The equal internal bisectors theorem, 1840 - 1940. ... Many solutions or none? A centenary account.
This paper contains (i) a short history of the geometrical theorem proposed in 1840 by Professor Daniel Christian Ludolph Lehmus (1780 - 1863) of Berlin to Jacob Steiner -
Incidentally two widely current legends will be cleared up, (i) that J J Sylvester in 1852 proved that a solution was impossible, (ii) that nevertheless a valid proof, the first, was given in 1874 by a Girl of the Golden West, a contemporary of Bret Harte and Mark Twain. Both these stories are "much exaggerated."
Steiner, like most mathematicians, found the theorem "very difficult," and Sylvester remarks, referring to J C Adams - "If report may be believed, intellects capable of extending the bounds of the planetary system, and lighting up new regions of the universe with the torch of analysis, have been baffled by the difficulties of the elementary problem under consideration." (Phil. Mag., 1853.)
Steiner gave a fine solution (Crelle's Journal, 1844), both for external and internal bisection, and found an external case where the theorem is not true. This occurs when BY and CZ meet AC produced and BA produced respectively.
Meanwhile in 1842 the Nouvelles Annales de Mathematiques of Paris proposed it for solution. Two proofs were given immediately, one by Charles Ernest Rougevin, a pupil at the College Louis le Grand, one by Grout de St Paer, a pupil at the Collège de Versailles.
From 1844 to 1852, about a dozen proofs appeared in Grunert's Archiv der Mathematik; in 1850-51 the theorem reached England. It was set in a Cambridge Examination Paper, with a new and disturbing element introduced. The proof was to be direct, i.e., without reductio ad absurdum.
This came to the notice of J J Sylvester, then writing on Equations and their Roots, and in the Philosophical Magazine for October 1852, he published two indirect proofs, one by B L Smith of Jesus College Cambridge.
He surmises (does not prove) that "when a theorem depends on the necessary non-existence of real roots (within prescribed limits) of the analytical equation expressing the conditions, no other form of proof then reductio ad absurdum is possible. If this is erroneous, it can be refuted in particular instances." But, he says, all proofs of the Bisectors Theorem have been hitherto indirect. He invited mathematicians to give a direct proof of the theorem -
I believe his proof is indirect. It is supported by Euclid III. 35, which uses the Theorem of Pythagoras, and by Euclid II. 6, involving the existence and construction of a square. This depends on Euclid I. 29, indirectly proved. J J Sylvester said nothing. The principle had been laid down definitely by him, that "all lemmas and supporting propositions must be provable directly." He did not admit that Rougevin had a right, if he claimed his proof as direct (which he did not) to assume that two triangles with equal vertical angles standing on the same side of the same base, have the same circumcircle. Few solvers have paid any attention to these just principles.
Rev Dr Adamson replied to Sylvester in three articles, admitting his Mathematics, but denying his logical deductions. He indicated what he thought might be a direct proof, but did not go fully into details.
He was followed by
Augustus De Morgan On Direct and Indirect Proofs. (London Edinburgh and Dublin Philosophical Magazine - December 1852.)
Any proof of the Contrapositive of a Proposition is a proof of the Positive, for their content is the same. Euclid, not writing for expert logicians, but for persons who through Geometry desired to become logicians, used reductio ad absurdum to pass from Contrapositive to Positive - quite unnecessarily.
De Morgan means that the Contrapositive and the Positive being identical logically, no further discussion is necessary if you have proved the Contrapositive. It has been lately pointed out to me by Mr J A Fullarton, ex-Headmaster, Ballymena Academy, that in Nixon's Euclid Revised and elsewhere it is proved that the bisector of the smaller of the two base angles of a triangle is longer than that of the other. This is the Contrapositive of our Theorem. But of course it has to be proved, and in doing so Nixon and others use supporting propositions only provable indirectly.
... we reject as indirect all proofs of the Bisectors theorem that depend on Euclid I. 32, which is proved indirectly by using Euclid I. 29. We must reject also proofs using the Theory of Proportion, or depending on the Theorem of Pythagoras, which ultimately depends on Euclid I. 29. (Transversal across two parallel lines makes interior angle-sum on one side equal to two right angles.) We have already rejected, as indirect, proofs of the Contrapositive, like Casey's, Steiner's, etc.
From 1852 to 1874 one finds in England alone about a dozen proof's - in the Lady's and Gentleman's Diary (devoted to Poetry and Mathematics), and in other journals. They are (1) frankly indirect, or (2) proofs of the Contrapositive, or (3) "direct" proofs depending on indirectly proved lemmas.
In 1874 arrived what is perhaps the best known solution. It was sent by Miss Christine Chart, Oakland, California, to Rev Dr N M Ferrers, Master of Gonville and Caius, who forwarded it with a covering letter to the Philosophical Magazine.
It had been made out in 1842, not by Miss Chart, but by her friend, Mr F G Hesse. Sylvester is said to have accepted it. I can hardly think he did, for it depends on Euclid I. 32, which depends, as shown above, on Euclid I. 29.
The best Contrapositive proof is attributed to John Casey by a writer in the Mathematical Gazette. It avoids Euclid I. 29 and 47, the Blue Symplegades on which many a solver's ship foundered.
Many proofs have appeared in the last 60 years. I am greatly indebted for valuable details to Mr J W Stewart, of Sunderland, formerly of Dumfries and Ayr. Also to an article by the late Dr J S Mackay in the Proceedings of the Edinburgh Mathematical Society.
I give Mr Stewart's fine proof, which is a concealed contrapositive. That is now no failing.
When this account was being written, in 1940, I wrote to Mr Stewart telling him that I was going to claim his as the only direct proof, provided he could give me a direct proof of Euclid I. 47. He at once sent the fine demonstration of the Theorem of Pythagoras attributed to Leonardo da Vinci. But, alas! a simple case of Euclid I. 14 is required in this, and the last hope was gone!
(i) More than 60 distinct proofs of the Theorem have been given, many frankly indirect.
(ii) Some of the best are proofs of the Contrapositive, i.e., indirect.
(iii) If it is held, as I hold, that Euclid I. 14, Euclid I. 29, Euclid I. 32, and the Theorem of Pythagoras have no direct proof, then the Bisectors Theorem has not been proved directly, nor is it likely to be.