Algebra.Com's Answer #818058 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "Triangle ABC has vertices A(-2,3), B(1,-3) and C(p,1).
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document.write( "For how many values of p is the triangle isosceles?
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document.write( " The solution by @MathLover1 is FATALLY INCOMPLETE.\r
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document.write( " There are several other, different isosceles triangles, that @MathLover1 missed/failed to find.\r
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document.write( " I came to bring the correct solution.\r
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document.write( "First, find the length of the side AB. It is (apply the distance formula)\r
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document.write( " d =  =  =  =  .\r
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document.write( "Next, consider different cases.\r
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document.write( "Case AB = BC.\r\n" );
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document.write( " In this case, BC = =  =  .\r\n" );
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document.write( " Squaring both sides, you get\r\n" );
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document.write( " 45 =  ,\r\n" );
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document.write( " 45 =  \r\n" );
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document.write( "  = 0\r\n" );
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document.write( "  =  =  =  .\r\n" );
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document.write( " Thus, it gives two different real values for p, and, hence, two different isosceles triangles with AB = BC.\r\n" );
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document.write( " These two different isosceles triangles are shown in the Figure 1 below (blue lines).\r\n" );
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document.write( "  \r\n" );
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document.write( " Figure 1. Two different triangles for case 1.\r\n" );
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document.write( "Case AB = AC.\r\n" );
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document.write( " In this case, AC = =  =  .\r\n" );
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document.write( " Squaring both sides, you get\r\n" );
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document.write( " 45 =  ,\r\n" );
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document.write( " 45 =  \r\n" );
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document.write( "  = 0\r\n" );
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document.write( " =  =  =  .\r\n" );
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document.write( " Thus, it gives two different real values for p, and, hence, two different isosceles triangles with AB = AC.\r\n" );
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document.write( " These two different isosceles triangles are shown in the Figure 2 below (blue lines).\r\n" );
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document.write( "  \r\n" );
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document.write( " Figure 2. Two different triangles for case 2.\r\n" );
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document.write( "So, in my post I shown 4 possible isosceles triangles, missed by @MathLover1.\r
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document.write( "The fifth triangle is the one found in her post.\r
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document.write( "The ANSWER to the problem's question IS : 5 triangles are possible.\r
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