document.write( "Question 405635: Prove that if an equilateral triangle is constructed on each side of a given triangle and the third vertex of each of these triangles is joined to the opposite vertex of the original triangle, then the three segments determined are congruent. \n" ); document.write( "

Algebra.Com's Answer #286568 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
Our diagram looks something like this:\r
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\n" ); document.write( "(It may appear that these three lines intersect at a point, but that might not necessarily be the case!)\r
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\n" ); document.write( "\n" ); document.write( "This may seem like an unusual solution, but I believe it is by far the easiest method, and more efficient than trying to brute-force the problem. I will begin by introducing complex numbers into the problem. We can assume that points A, B, ..., F are points on the complex plane.\r
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\n" ); document.write( "\n" ); document.write( "Suppose that A is the origin of the complex plane, i.e. A = 0, and that B and C are arbitrary complex numbers. It follows that:\r
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\n" ); document.write( "\n" ); document.write( " $\"F+=+C%2Aomega\"$ and $\"B+=+D%2Aomega\"$ where

(The Greek letter \"omega\" denotes a root of unity, in this case, a 6th root of unity. These equations basically state that F is equal to point C rotated about $\"pi%2F6\"$ , and B is the same as D rotated about $\"pi%2F6\"$ .) We want to show that CD and BF have equal magnitudes. Since CD = D - C and BF = F - B (this is similar to subtracting vectors), then we want to show that\r
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\n" ); document.write( "|D - C| = |F - B| where |z| denotes the magnitude of the complex number z.\r
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\n" ); document.write( "\n" ); document.write( "Note that $\"F+=+C%2Aomega\"$ and $\"B+=+D%2Aomega\"$ . We can substitute these into our equation to obtain\r
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\n" ); document.write( "\n" ); document.write( "|D - C| = |(C - D)*omega|. Here, we note that the magnitude of D-C is the same as C-D (i.e. a complex number in the form a+bi has the same magnitude as the number -a-bi). Also, the magnitude of $\"omega\"$ is equal to $\"sqrt%28%281%2F2%29%5E2+%2B+%28sqrt%283%29%2F2%29%5E2%29\"$ which is 1. Hence, the magnitude remains unchanged, and we conclude that |D - C| = |F - B| and that CD = BF. By a simple symmetry argument (e.g. assigning B as the origin to show that BF = AE), we conclude that CD = BF = AE, and we are done.\r
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