document.write( "Question 1181779: Prove that the sum of the three altitudes of a triangle is less than the sum of three sides of a triangle. \n" ); document.write( "

Algebra.Com's Answer #811722 by ikleyn(52908) $\"\"$ $\"About$

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\n" ); document.write( "Prove that the sum of the three altitudes of a triangle is less than the sum of three sides of a triangle.
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document.write( "Let \"a\", \"b\" and \"c\"  be  the lengths of the three sides of the triangle.\r\n" );
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document.write( "Let f, g and h  be  the altitudes of the triangle, drawn to the sides \"a\", \"b\" and \"c\", respectively.\r\n" );
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document.write( "Then we have\r\n" );
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document.write( "    f <= b  and  f <= c;                            (1)\r\n" );
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document.write( "    so  2f <= b+c,  which means  f <= .       (2)\r\n" );
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document.write( "Notice that of the two inequalities (1), at least one inequality is the STRICT inequality;\r\n" );
document.write( "therefore, the final inequality (2) is the STRICT inequality, too,\r\n" );
document.write( "so we can write instead of (2) strict inequality\r\n" );
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document.write( "        f < .                                  (3)\r\n" );
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document.write( "Similarly to inequality (3), we can prove the following two inequalities\r\n" );
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document.write( "        g < ,                                  (4)\r\n" );
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document.write( "        h < .                                  (5)\r\n" );
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document.write( "Now add inequalities  (3), (4) and (5).  You will get\r\n" );
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document.write( "        f + g + h < ,\r\n" );
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document.write( "or\r\n" );
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document.write( "        f + g + h < a + b + c.\r\n" );
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document.write( "It is exactly what should be proved.\r\n" );
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