000RM.dvi

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Pi Mu Epsilon Journal, Problem 1058, proposed by P. A. Lindstrom,
Batavia, NY

Suppose that triangleABChas an interior pointP. LetD,E,F be
points on sidesAB,BC,CArespectively, so thatPD⊥AB,PE⊥
BC,PF⊥ CA. Let|AB|= x,|BC| = y,|CA|= z,|AB| =a,
|BE|=b, and|CF|=c.

1.Show that(x−a)^2 +(y−b)^2 +(z−c)^2 =a^2 +b^2 +c^2.

2.Show that if ABCis an equilateral triangle, thena+b+c=

1

2 (perimeter) of triangleABC.