document.write( "Question 419848: Prove that the sum of the distances from any point in the interior of a equilateral triangle to each of the sides of the triangle is equal to the length of an altitude of that triangle. \n" ); document.write( "
Algebra.Com's Answer #293506 by richard1234(7193)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "It's a little hard to draw here, but let x,y,z be the altitudes from D onto AB, BC, CA, respectively (segment x is perpendicular to AB, etc.). We know that the sum of the areas of the triangles ADB, BDC, CDA add up to the area of ABC, so\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%2AAB%29%2F2+%2B+%28y%2ABC%29%2F2+=+%28z%2ACA%29%2F2+=+%28h%2AAB%29%2F2\" where h is the altitude from C onto AB.\r
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\n" ); document.write( "\n" ); document.write( "Since the triangle is equilateral, \"AB+=+BC+=+CA\" and we can replace BC, CA with AB, without loss of generality.\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%2AAB%29%2F2+%2B+%28y%2AAB%29%2F2+%2B+%28z%2AAB%29%2F2+=+%28h%2AAB%29%2F2\"\r
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\n" ); document.write( "\n" ); document.write( "We can multiply both sides by \"2%2FAB\", cancelling out the AB and 2. Hence, we obtain\r
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\n" ); document.write( "\n" ); document.write( "\"x%2By%2Bz+=+h\", as desired.
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