document.write( "Question 1120393: Can there exist a triangle ROS in which the trisectors of angle O intersect RS at D and E with RD =1,,DE=2,andES=4? Explain \r
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Algebra.Com's Answer #810695 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "Make a sketch.  I will refer to it as if you have it in front of you (as I have it in front of my eyes).\r\n" );
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document.write( "    |    Let ROS be a triangle, and let its angle ROS at vertex O   |\r\n" );
document.write( "    |    is trisected in a way that RD= 1, DE=2 and ES=4.           |    (*)\r\n" );
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document.write( "I am going to bring it to contradiction and prove, in this way, that such trisection IS NOT POSSIBLE.\r\n" );
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document.write( "Let \"t\" be any of the three congruent small angles at vertex O.\r\n" );
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document.write( "Let \"a\" be the length of RO;  let \"b\" be the length of DO.\r\n" );
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document.write( "Then  the length of  EO  is  2a  (bisector theorem for triangle REO), \r\n" );
document.write( "and   the length of  SO  is  2b  (bisector theorem for triangle DOS).\r\n" );
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document.write( "    I will remind you, that the bisector theorem STATES that\r\n" );
document.write( "    the ratio of the segments lengths, to which the bisector of a triangle\r\n" );
document.write( "    divides the opposite side is equal to the ratio of the lengths\r\n" );
document.write( "    of the two lateral corresponding sides.\r\n" );
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document.write( "Now,  triangle ROD has side lengths \"a\" and \"b\", that conclude the angle \"t\", and the opposite side RD= 1.\r\n" );
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document.write( "So,  we write the Cosine Law formula for RD\r\n" );
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document.write( "     RD^2 =  = 1^2.                  (1)\r\n" );
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document.write( "Next, triangle DOE has side lengths \"b\" and \"2a\", that conclude the angle \"t\", and the opposite side DE= 2.\r\n" );
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document.write( "So, we write the Cosine Law formula for DE\r\n" );
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document.write( "    DE^2 =  = 2^2.            (2)\r\n" );
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document.write( "Finally, triangle EOS has side lengths \"2a\" and \"2b\", that conclude the angle \"t\", and the opposite side ES= 4.\r\n" );
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document.write( "So, we write the Cosine Law formula for ES\r\n" );
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document.write( "    ES^2 =  = 4^2.      (3)\r\n" );
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document.write( "I will rewrite the three equations (1), (2) (3) in simpler form\r\n" );
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document.write( "             = 1.                   (1')\r\n" );
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document.write( "             = 4.                  (2')\r\n" );
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document.write( "             = 16.                (3')\r\n" );
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document.write( "Now compare equations  (1')  and  (3').   You see that the left side of (3') is four times the left side of (1'),\r\n" );
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document.write( "but the right side of (3') is sixteen times left side of (1').\r\n" );
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document.write( "It is the contradiction, and this contradiction proves that the original assumption (*) is not valid / (is not possible).\r\n" );
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\n" ); document.write( "\n" ); document.write( "At this point, my proof is completed.\r
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\n" ); document.write( "\n" ); document.write( "It is also possible to construct another proof, using the similarity properties of small triangles.\r
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document.write( "    Indeed, from one hand side, triangles ROD and EOS must be similar with the similarity coefficient of 2,\r\n" );
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document.write( "    because their \"lateral\" sides are proportional  RO : DO = a : b = EO : SO = (2a) : (2b) and the concluded angles \"t\" are congruent;\r\n" );
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document.write( "    but from the other hand side,  their sides, opposite to angle \"t\", are with the ratio 4:1, which gives the CONTRADICTION.\r\n" );
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