document.write( "Question 1202365: Points A, B, and C are given in the coordinate plane. There exists a point Q and a constant k such that for any point P,\r
\n" ); document.write( "\n" ); document.write( "PA^2 + PB^2 + PC^2 = 3PQ^2 + k.\r
\n" ); document.write( "\n" ); document.write( "If A = (4,-1), B = (-3,1), and C = (5,-3), then find the constant k.\r
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Algebra.Com's Answer #837129 by ikleyn(52810) $\"\"$ $\"About$

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document.write( "Let P be (x,y).  Then\r\n" );
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document.write( "PA^2 + PB^2 + PC^2 = \r\n" );
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document.write( "= (x-4)^2 + (y+1)^2 + (x+3)^2 + (y-1)^2 + (x-5)^2 + (y+3)^2 =\r\n" );
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document.write( "= 3x^2 + 3y^2 - 12x + 6y + 61 = \r\n" );
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document.write( "=3(x^2 + y^2 - 4x + 2y) + 61 =\r\n" );
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document.write( "= 3((x-2)^2 + (y+1)^2) + 46.\r\n" );
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document.write( "This shows that if  Q = (2,-1),  then  PQ^2 = (x-2)^2 + (y+1)^2.\r\n" );
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document.write( "And so  PA^2 + PB^2 + PC^2 = 3PQ^2 + 46.\r\n" );
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document.write( "ANSWER.  k = 46.\r\n" );
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