Quadro Phobia

Geometry Level 4

If a , b , c a , b ,c and d d are the sides of a quadrilateral then find the infimum of ( a 2 + b 2 + c 2 ) / d 2 . (a^2+b^2+c^2)/d^2 .


The answer is 0.333.

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1 solution

Shandy Rianto
Jul 9, 2015

Let a = b = c = x a = b = c =x

To find the minimum value of ( a 2 + b 2 + c 2 ) / d 2 (a^2+b^2+c^2)/d^2 we should maximize d d

The maximum value of d d should not be greater than the total of a a , b b and c c so d ≤ 3 x d \leq 3x

a 2 + b 2 + c 2 d 2 \frac{a^2+b^2+c^2}{d^2}

≥ x 2 + x 2 + x 2 ( 3 x ) 2 \geq \frac{x^2+x^2+x^2}{(3x)^2}

≥ 3 x 2 9 x 2 \geq \frac{3x^2}{9x^2}

≥ 1 3 \geq\frac{1}{3}

≥ 0.333 \geq \boxed{0.333}

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