What is equilateral: Definition and 2 Discussions

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

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  1. Hill

    Triangle of centroids

    Here is my sketch: The triangle ##abc## is arbitrary, the triangles ##acp##, ##abq##, and ##bcr## are equilateral with centroids ##m##, ##n##, and ##k##. I suspect that the triangle ##mnk## is equilateral. Here is my proof. By the equation for centroids, ##3m=a+c+p## ##3n=a+q+b## ##3k=b+c+r##...
  2. Hill

    Three points on the Complex plane

    ##arg((b-a)/(c-a))## is an angle between ##ab## and ##ac##. ##arg((a-c)/(b-c))## is an angle between ##ca## and ##cb##. For them to be equal, ##b## has to be equidistant from ##a## and ##c##, i.e. ##|b-a|=|b-c|##. Then the equation for distances becomes, ##|b-a|/|c-a|=|c-a|/|b-a|##. Thus...
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