When algebra meets geometry

Geometry Level 5

Let a a , b b and c c be the lenghts of the sides of a triangle. Supose that they satisfy a 2 + b 2 = k c 2 a^2+b^2=kc^2 . If k > M k>M find the maximum value of M M


The answer is 0.5.

Sergi Mlg Sabat by Sergi Mlg Sabat , 29, USA

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1 solution

( 1 − k ) c 2 < 2 a b ≤ a 2 + b 2 = k c 2 ⇒ 1 < 2 k ⇒ k > 0.5 (1 - k)c^2 < 2ab \leq a^2 + b^2 = kc^2 \Rightarrow 1 < 2k \Rightarrow k > 0.5 .

Details: Using AM -GM (or simply a 2 + b 2 − 2 a b ≥ 0 a^2 + b^2 - 2ab \ge 0 ) inequality, we get 2 a b ≤ a 2 + b 2 = k c 2 2ab \leq a^2 + b^2 = k c^2 .

and due to triangular inequality. 0 < c < a + b ⇒ c 2 < a 2 + b 2 + 2 a b = k c 2 + 2 a b ⇒ ( 1 − k ) c 2 < 2 a b 0 < c < a + b \Rightarrow c^2 < a^2 + b^2 + 2ab = kc^2 + 2ab \Rightarrow (1 - k)c^2 < 2ab . So the maximum value for M M is 0.5

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