If ( x , y , z) be the lengths of perpendiculars from any interior point P of a triangle A B C on sides B C, C A and A B respectively then find the minimum value of : x 2 + y 2 + z 2 The sides of triangle being a , b , c.I thought of using Lagrange's method of multipliers but am not able to find another function in terms of x , y , z and a , b , cAny help will be appreciated. Thanks.

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If (  x  ,  y  ,  z) be the lengths of perpendiculars from any interior point P of a triangle   A  B  C on sides   B  C,  C  A and   A  B respectively then find the minimum value of :      x    2    +      y    2    +      z    2  The sides of triangle being   a  ,  b  ,  c.I thought of using Lagrange&#039;s method of multipliers but am not able to find another function in terms of   x  ,  y  ,  z and   a  ,  b  ,  cAny help will be appreciated. Thanks.

Esteban Johnson · Accepted Answer

Here is a solution without the method of Lagrange multipliers. Note that   a  x  +  b  y  +  c  z  =  2  S where   S is the area of the triangle. Therefore, by Cauchy-Schwarz inequality we have            4              S        2                            a        2            +              b        2            +              c        2              =            (      a      x      +      b      y      +      c      z              )        2                            a        2            +              b        2            +              c        2              ≤      x    2    +      y    2    +      z    2    .The equality holds when   x  :  y  :  z  =  a  :  b  :  c. The corresponding point is the radical center of the three Apollonian circles associated with the triangle.

Quintin Stafford · Accepted Answer

As in       x    2    +      y    2    +      z    2   all are squares , sum of squares is minimum when all terms (for which squares are taken) are equal. So the point should be the orthocentre of triangle with equal altitudes . if altitude be &#039;h&#039;. distance from orthocentre to base is h/2 , then minimum is   3  (      h    2    )      /    4. height can be expresses in terms of area which is in terms of sides . so a relation is obtained as you wanted .

If ( x , y , z ) be the lengths of perpendiculars from any interior point P of a tri

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