perplexus.info :: Geometry : Within a triangle

Within a triangle (Posted on 2018-10-13)

Prove: If the sides a, b, c of a triangle satisfy a^2 + b^2 = kc^2, then k >1/2.

No Solution Yet Submitted by Ady TZIDON

Rating: 3.0000 (1 votes)

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Proof (spoiler)

Comment 1 of 1

Because this is a triangle,

(1) a+b > c,

Squaring gives

(2) a^2 + b^2 + 2ab > c^2

For all numbers

(3) (a-b)^2 >= 0

Expanding gives

(4) a^2 + b^2 - 2ab >= 0

Adding 2 and 4 gives

(5) 2(a^2 + b^2) > c^2

Dividing by 2 gives

(6) a^2 + b^2 > (1/2) * c^2

q.e.d.

Posted by Steve Herman on 2018-10-13 09:40:24

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