document.write( "Question 1119540: Find the smallest possible value:\r
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Algebra.Com's Answer #735141 by ikleyn(52776) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "The key to the solution of the problem is to recognize that the given expression is the sum of distances from the point (x,y)
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\n" ); document.write( "\n" ); document.write( "Theorem\r
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\n" ); document.write( "\n" ); document.write( " For a convex quadrilateral in a plane, the point in the plane which minimizes the sum of the distances from the point to vertices
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document.write( "          Similar statement for a triangle leads to Fermat's point of a triangle and is considered as a difficult geometry\r\n" );
document.write( "          conception and statement, which goes far beyond and above the elementary geometry level.\r\n" );
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document.write( "              See this Wikipedia article  https://en.wikipedia.org/wiki/Fermat_point .\r\n" );
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document.write( "          But for a quadrilateral it is ELEMENTARY statement accesible and approachable for starters.\r\n" );
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document.write( "Let ABCD be the given quadrilatersl in a plane with the verices A, B, C and D  (in this order).\r\n" );
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document.write( "Let \"O\" be the intersection point of its diagonals AC and BD.\r\n" );
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document.write( "And let X be any other point in the plane. \r\n" );
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document.write( "The sum of distances from X to vertices is \r\n" );
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document.write( "d(X) = |AX| + |BX| + |CX| + |DX|.\r\n" );
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document.write( "The sum of distances from O to vertices is \r\n" );
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document.write( "d(O) = |AO| + |BO| + |CO| + |DO|.\r\n" );
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document.write( "By applying the \"triangle inequality\", you have\r\n" );
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document.write( "d(O) = (|AO| + |CO|) + (|BO| + |DO|) = |AC| + |BD| < (|AX| + |CX|) + (|BX| + |DX|) = d(X),\r\n" );
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document.write( "and the statement is PROVED.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Therefore, the solution to your problem is THIS:\r
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document.write( "    The point in the plane which gives the minimum to your expression is the intersection point of the segment\r\n" );
document.write( "    connecting the points A=(0,0) and C=(3,4) with the segment connecting the points B=(0,1) and D=(1,0).\r\n" );
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document.write( "The straight line connecting the points A and C is\r\n" );
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document.write( "    y = .     (1)\r\n" );
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document.write( "The straight line connecting the points B and D is\r\n" );
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document.write( "    y - 1 = -x.      (2)\r\n" );
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document.write( "Their intersection is the point \r\n" );
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document.write( "    = 1 - x  ====>  4x = 3 - 3x  ====>  7x = 3  ====>  x = ;  y =  = .\r\n" );
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document.write( "To find the minimum of the given expression, you need to find the lengths of the diagonals  |AC| and |BD|  and add them.\r\n" );
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document.write( "|AC| =  =  =  = 5;\r\n" );
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document.write( "|BD| = .\r\n" );
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document.write( "So, the minimum of the given expression is  .\r\n" );
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document.write( "Answer.  The point which gives the minimum to the given expression is (x,y) = (,).\r\n" );
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document.write( "         The value of the minimum is .\r\n" );
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