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Question
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
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Solution
Let A (1, 2, 3) , B (2, 3, 1) and C (3, 1, 2) are the coordinates of the triangle \[\bigtriangleup ABC\]
AB =\[\sqrt{\left( 2 - 1 \right)^2 + \left( 3 - 2 \right)^2 + \left( 1 - 3 \right)^2}\]
\[= \sqrt{\left( 1 \right)^2 + \left( 1 \right)^2 + \left( - 2 \right)^2}\]
\[ = \sqrt{1 + 1 + 4}\]
\[ = \sqrt{6}\]
BC =\[\sqrt{\left( 3 - 2 \right)^2 + \left( 1 - 3 \right)^2 + \left( 2 - 1 \right)^2}\]
\[ = \sqrt{1 + 4 + 1}\]
\[ = \sqrt{6}\]
\[ = \sqrt{4 + 1 + 1}\]
\[ = \sqrt{6}\]
Therefore, it is an equilateral triangle.
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