A geometric inequality in a triangleIf x , y , z &gt; 0 , could it be: z ( 2 z + x ) z + x b 2 + y ( 2 y + x ) y + x c 2 &gt; y z y + z a 2 for a triangle A B C , with a = B C , b = A C , c = A B ?If b 2 + c 2 = a 2 , then the inequality holds, since z ( 2 z + x ) z + x − y z y + z = z 2 ( x + y + 2 z ) ( z + x ) ( y + z ) , y ( 2 y + x ) y + x − y z y + z = y 2 ( x + 2 y + z ) ( z + x ) ( y + z ) If b 2 + c 2 &gt; a 2 , then the inequality holds, since z ( 2 z + x ) z + x b 2 + y ( 2 y + x ) y + x c 2 &gt; y z y + z b 2 + y z y + z c 2 &gt; y z y + z a 2 I can’t handle the case b 2 + c 2 &lt; a 2 and I need some help

Question

A geometric inequality in a triangleIf   x  ,  y  ,  z  &amp;gt;  0  , could it be:            z      (      2      z      +      x      )              z      +      x            b    2    +            y      (      2      y      +      x      )              y      +      x            c    2    &amp;gt;            y      z              y      +      z            a    2  for a triangle   A  B  C  , with   a  =  B  C  ,  b  =  A  C  ,  c  =  A  B  ?If       b    2    +      c    2    =      a    2  , then the inequality holds, since            z      (      2      z      +      x      )              z      +      x        −            y      z              y      +      z        =                    z        2            (      x      +      y      +      2      z      )              (      z      +      x      )      (      y      +      z      )        ,            y      (      2      y      +      x      )              y      +      x        −            y      z              y      +      z        =                    y        2            (      x      +      2      y      +      z      )              (      z      +      x      )      (      y      +      z      )      If       b    2    +      c    2    &amp;gt;      a    2  , then the inequality holds, since            z      (      2      z      +      x      )              z      +      x            b    2    +            y      (      2      y      +      x      )              y      +      x            c    2    &amp;gt;            y      z              y      +      z            b    2    +            y      z              y      +      z            c    2    &amp;gt;            y      z              y      +      z            a    2  I can’t handle the case       b    2    +      c    2    &amp;lt;      a    2   and I need some help

Emma Singleton · Accepted Answer

By C-S            z      (      2      z      +      x      )              z      +      x            b    2    +            y      (      2      y      +      x      )              y      +      x            c    2    ≥            (      b      +      c              )        2                                      z          +          x                          z          (          2          z          +          x          )                    +                        y          +          x                          y          (          2          y          +          x          )                      &amp;gt;  &amp;gt;            a      2                                z          +          x                          z          (          2          z          +          x          )                    +                        y          +          x                          y          (          2          y          +          x          )                      &amp;gt;            a      2                                z          +          x                          z          (          z          +          x          )                    +                        y          +          x                          y          (          y          +          x          )                      =                    a        2            y      z              y      +      z        .

A geometric inequality in a triangle If x,y,z>0, could it be: ( z(2z+x))/(z+x) b^2+( y(2y+x) )/(y+x ) c^2> (yz)/(y+z) a^2 for a triangle ABC, with a=BC, b=AC, c=AB?

Answered question

Answer & Explanation

New Questions in Pre-Algebra