Here is Legendre’s lemma—which is needed for his proof found in many texts, base

Question

Here is Legendre’s lemma—which is needed for his proof found in many texts, based on the Archimedean property of angles—that the angle sum of every triangle is <=180 . He got the idea for this from Euclid’s construction in his (incomplete) proof of the exterior angle theorem (I.16). Given triangle ABC. Let D be the midpoint of BC. Let E be the point on the ray opposite to DA such that DE congruent DA. Prove that triangle AEC has the same angle sum as triangle ABC and that either angle AEC or angle EAC is less than or equal ½(angle BAC). Use Legendre’s lemma to give RAA proof of the Saccheri-legendre theorem

Explanation / Answer

BA to D , and make AD equal to AC . Join CD. Then because AD is equal to AC, the angle ACD is equal to ADC ; therefore the angle BCD is greater than the angle BDC; hence the side BD opposite to the greater angle is greater than BC opposite to the less . Again, since AC is equal to AD, adding BA to both, we have the sum of the sides BA, AC equal to BD. Therefore the sum of BA, AC is greater than BC. Or thus:

Bisect the angle BAC by AE Then the angle BEA is greater than EAC; but EAC = EAB (const.); therefore the angle BEA is greater than EAB. Hence AB is greater than BE . In like manner AC is greater than EC. Therefore the sum of BA, AC is greater than BC.

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