True or false? Any triangle we can h A 2 + h B 2 ≥ 2 h C 2 , where h A , h B , h C are lengths of the heights from vertices A, B and CLet △ A B C any triangle and the lengths of the heights from vertices A, B and C are, respectively, h A , h B , h C . Then how should we prove that, h A 2 + h B 2 ≥ 2 h C 2 , h C 2 + h A 2 ≥ 2 h B 2 a n d h B 2 + h C 2 ≥ 2 h A 2

Question

True or false? Any triangle we can       h    A    2    +      h    B    2    ≥  2      h    C    2  , where       h    A    ,      h    B    ,      h    C   are lengths of the heights from vertices A, B and CLet   △  A  B  C any triangle and the lengths of the heights from vertices A, B and C are, respectively,       h    A  ,       h    B  ,       h    C  . Then how should we prove that,       h    A    2    +      h    B    2    ≥  2      h    C    2    ,        h    C    2    +      h    A    2    ≥  2      h    B    2      a  n  d        h    B    2    +      h    C    2    ≥  2      h    A    2

Joel Reese · Accepted Answer

Step 1We may use following inequality:      h    A    +      h    b    +      h    c    &amp;lt;  a  +  b  +  c  =  2  p                 (  1  )Also:      h    A    +      h    B    +      h    C    &amp;gt;  pLet&#039;s suppose:      h    A    +      h    B    +      h    C    ≈  p  +      p    2    =            3      p        2  Squaring both sides we get:      h    A    2    +      h    B    2    =      h    C    2    +      9    4        p    2    −  3  p  ⋅      h    C    −  2      h    A        h    B  So we must show that:  t  =      9    4        p    2    −  3  p  ⋅      h    C    −  2      h    A        h    B    ≥      h    C    2  Step 2If       h    A    =      h    B    =  ϵ then   p  ≈      h    C   and we have:  t  =      9    4    ×  4      h    C    2    −  3      h    C    2    −  ϵ  =  6      h    C    2    −  ϵSo there is possibility to have a triangle in which:      h    A    2    +      h    B    2    ≥      h    C    2  But there is no guarantee for this relation to be symmetric. for example Consider Right angle triangle with sides 3, 4, 5 where:      h    B    =  4,       h    C    =  3 and       h    A    =  2.4 we have:      h    A    2    +      h    B    2    =      2.4    2    +      4    2    =  21.76  &amp;gt;  2  ×      3    2    =  18      h    B    2    +      h    C    2    =  9  +  16  =  25  &amp;gt;  2  ×      2.4    2    =  11.54But:       h    A    2    +      h    C    2    =  14.76  &amp;lt;  2  ×      4    2    =  32

jhenezhubby01ff · Accepted Answer

Explanation:Consider a very narrow pointy isosceles triangle, with one angle close to       0    ∘   and the others nearly       90    ∘  .

True or false? Any triangle we can h_{A}^{2}+h_{B}^{2} geq 2h_{C}^{2}, where h_A, h_B, h_C are lengths of the heights from vertices A, B and C.

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