Condition for squaring the sides of a triangle

Geometry Level 3

Triangle ABC has side lengths a , b , c a,b,c .

In order for a 2 , b 2 , c 2 a^{2}, b^{2},c^{2} to be the side lengths of a triangle, what is the necessary and sufficient condition for the angles of triangle ABC?

Triangle ABC is acute Triangle ABC is right-angled Triangle ABC is obtuse Impossible

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Irina Stanciu by Irina Stanciu , 21, United Kingdom

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1 solution

Irina Stanciu
Dec 27, 2016

The positive real numbers a 2 , b 2 , c 2 a^{2}, b^{2},c^{2} can be the side lengths if and only if
a 2 + b 2 > c 2 \ a^{2}+ b^{2}> c^{2} , b 2 + c 2 > a 2 b^{2}+ c^{2}> a^{2} , a 2 + c 2 > a 2 a^{2}+ c^{2}> a^{2} . \quad\quad\quad (1) Because
cos ⁡ A \cos A = b 2 + c 2 − a 2 2 b c \frac{b^{2}+c^{2}-a^{2}}{2bc}
c o s B cos B = c 2 + a 2 − b 2 2 c a \frac{c^{2}+a^{2}-b^{2}}{2ca}
c o s C cos C = a 2 + b 2 − c 2 2 a b \frac{a^{2}+b^{2}-c^{2}}{2ab}

Relation (1) is equivalent to c o s A > 0 , c o s B > 0 , c o s C > 0. cos A>0 , cos B>0, cos C>0. Hence the necessary and sufficient condition is that triangle ABC is acute .

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