1.7. Inequalities in triangles, Kiselev's Geometry, Book I. Planimetry, Givental Alexander.

 A Memorable Theorem.

Theorem: In a triangle, each side is smaller than the sum of the other two sides.

Proof: We have the triangle ABC and we can see that AB is clearly smaller than the sum of sides BC and AC. Also, that side BC is smaller than the sum of AB and AC is clear. But that AC is smaller than the sum of AB and BC is not so clear.

We will prove the last statement. In triangle ABC extend AB to AD such that BD and BC are congruent. This implies that triangle BCD is isosceles. So ∠BCD = ∠BDC. Now ∠ACD ＞∠BCD, i.e. ∠ACD＞∠BDC. This implies that side AD is greater than AC, in other words, AD = AB + BD ＞AC. Or AB + BC＞AC. This proves the last statement. 

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