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Math Help - minimizing squares...

  1. #1
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    minimizing squares...

    Given a triangle \triangle ABC and a straight line l, find the point P on l such that (PA)^2 + (PB)^2 + (PC)^2 is the smallest.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello,
    Quote Originally Posted by scipa View Post
    Given a triangle \triangle ABC and a straight line l, find the point P on l such that (PA)^2 + (PB)^2 + (PC)^2 is the smallest.
    Let A_h,\,B_h and C_h be the orthogonal projection of A,\,B and C on l, respectively. Using Pythagorean theorem :

    \begin{cases}<br />
PA^2=PA_h^2+AA_h^2\\<br />
PB^2=PB_h^2+BB_h^2\\<br />
PC^2=PC_h^2+CC_h^2\\<br />
\end{cases}

    hence

    PA^2+PB^2+PC^2=\underbrace{AA_h^2+BB_h^2+CC_h^2}_{  \text{that's a constant}}+PA_h^2+PB_h^2+PC_h^2<br />

    As AA_h^2+BB_h^2+CC_h^2 doesn't depend on P and PA_h^2+PB_h^2+PC_h^2\geq 0, minimizing PA^2+PB^2+PC^2 is equivalent to minimizing PA_h^2+PB_h^2+PC_h^2. How can this be achieved ?
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