Compute the least value of f(p) - ISL 1986

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Compute the least value of f(p) - ISL 1986

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#1 h Aug 31, 2010, 10:30 AM • 1 Y

Y by Adventure10

A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$ . Let $f(P) = AP + BP + CP + DP$ , where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

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#2 h Aug 23, 2012, 9:14 AM • 3 Y

Y by Amir Hossein, Adventure10, Mango247

1). If $|AB|=|CD|=c;\, |AC|=|BD|=b,\,|AD|=|BC|=a$ then, $ABCD$ isoscele tetrahedron.

2).
$f(P)=|PA|+|PB|+|PC|+|PD|=min\Leftrightarrow \{\begin{array}{c} \angle{APB}\equiv\angle{CPD}\\ \angle{APC}\equiv\angle{BPD}\\ \angle{APD}\equiv\angle{BPC}.\end{array}$
$1) \mbox{ and } 2)\Rightarrow f(P)=min\Leftrightarrow P=G(barycenter)=O(circumcenter)\Leftrightarrow \\ \Leftrightarrow |PA|=|PB|=|PC|=|PD|=R (circumradius)\Rightarrow \\ \Rightarrow f(P)=4R=\sqrt{2(a^2+b^2+c^2)}.$

This post has been edited 1 time. Last edited by mihai miculita, Aug 24, 2012, 5:08 AM

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#3 h Aug 24, 2012, 5:02 AM • 3 Y

Y by Amir Hossein, Adventure10, Mango247

$\mbox{Let }M\mbox{-the midpoint of }[AB] \mbox{ and } N\mbox{-midpoint of }[CD]; \mbox{ then: }MN\perp AB;CD.$
$\mbox{In general if }|AB|\neq|CD|,\mbox{ then: } \boxed{f(P)=\mbox{minim, for the point }P_0\in[MN]; \dfrac{|P_0M|}{|P_0N|}=\dfrac{|AB|}{|CD|}\neq 1}.$

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