document.write( "Question 413128: Prove that the sum of distances from any point in the interior of a triangle to the three vertices a, b, and c is greater than half the perimeter of the triangle. \n" ); document.write( "

Algebra.Com's Answer #290053 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "\r\n" );
document.write( "Let the sides of the large triangle be a, b, and c.\r\n" );
document.write( "Let the distances to the vertices from an interior point \r\n" );
document.write( "be x, y, and z.\r\n" );
document.write( "\r\n" );
document.write( "Half the perimeter of the large triangle is \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Use the triangular inequality on each of the three smaller triangles \r\n" );
document.write( "that make up the large triangle:\r\n" );
document.write( "\r\n" );
document.write( "x + y     > a\r\n" );
document.write( "x     + z > b\r\n" );
document.write( "    y + z > c\r\n" );
document.write( "\r\n" );
document.write( "Adding those three inequalities term by term:\r\n" );
document.write( "\r\n" );
document.write( " x +  y      > a\r\n" );
document.write( " x      +  z > b\r\n" );
document.write( "      y +  z > c\r\n" );
document.write( "--------------------\r\n" );
document.write( "2x + 2y + 2z > a+b+c\r\n" );
document.write( "\r\n" );
document.write( "Dividing through by 2\r\n" );
document.write( "\r\n" );
document.write( " x +  y +  z >  \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );