The geometric theorem “in an isosceles triangle the bisectors from the base angles are equal” is very simple to prove. Simple congruence of triangles and you’re done. Lower secondary education…
But the inverse statement “if the bisectors from the base angles are equal, then the triangle is isosceles” is a very different story. I fail to prove this seemingly equally simple proposition. Can anyone provide the proof? And more importantly, why is this such a difficult task?
Answer
For the sake of simplicity I mean by A the statement “The triangle is isosceles” and by B the statement “Two bisectors are equal”. There is no reason to believe that there should be any correlation between the simplicity or complexity of the proof of the statement “If A, then B” on the one hand and the proof of the statement “If B, then A”. ” on the other hand. One proof can be very simple and the other very complex. As for the specific problem, the website http://www.pandd.demon.nl/steileh.htm offers a whole host of proofs, some more ingenious than others and some that can still be considered relatively simple. consider.
Answered by
prof.dr. Jean Paul VAN BENDEGEM
logic, philosophy of mathematics and philosophy of science
Avenue de la Plein 2 1050 Ixelles
http://www.vub.ac.be/
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