What term/identity/theorem states that given a triangle the largest angle must be opposite the longest side?

Question

Calvin Oneill · Accepted Answer

Step 1It is a result of the law of sines:            sin      ⁡      A        a    =            sin      ⁡      B        b    =            sin      ⁡      C        c  This means tha the largest of a,b,c corresponds to the largest of   sin  ⁡  A  ,  sin  ⁡  B  ,  sin  ⁡  C  . .Assume A is the largest angle.If   A  ≤      π    2    , , then, since sinx is increasing on   [  0  ,  π      /    2  ]  , ,   sin  ⁡  A is the largest of the sines, so a is largest.If   A  &amp;gt;      π    2    , , then   sin  ⁡  (  A  )  =  sin  ⁡  (  π  −  (  B  +  C  )  )  =  sin  ⁡  (  B  +  C  )  , , with   0  ≤  B  ,  C  ≤  B  +  C  &amp;lt;      π    2    .. So   sin  ⁡  A  ≥  sin  ⁡  B  ,  sin  ⁡  C  ,  and again sinA is the largest value, and hence a is the longest length.Step 2Alternatively, more geometrically, find the circle which goes through A, B, C, and then, at least for A, B, C acute angles, the lengths of the sides and the lengths of the corresponding arc lengths, and the arc lengths correspond to twice the opposite angle.You then have to deal with the obtuse case separately. Not hard to do - type get the length corresponds to the arc of   2  π  −  2  A  =  2  B  +  2  C  &amp;gt;  2  B  ,  2  C and thus a is still longer then both b, c.

revistasbibxjm87 · Accepted Answer

Such result is known as:&quot;The side of a triangle opposite the largest angle is the longest side.&quot;You can find its proof in any plane Euclidean geometry book.

What term/identity/theorem states that given a triangle the largest angle

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