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Not only do we know that the four points lie on a circle, we also know their order around the circle: circumcentre, nine-point centre, first Fermat point and second Fermat point (C-N-F-S), either clockwise or counterclockwise. Which case occurs when depends only on the shape of the triangle and not on its size. In the applet below, the size of the triangle is fixed by fixing two of its vertices (black); the third vertex can be dragged to produce a triangle of any shape. |
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When the third vertex is in the blue region, the order C-N-F-S is clockwise, and when it is in the pink region, the order is counterclockwise. When it is on one of three boundaries of these regions - the circles or the vertical line - the triangle becomes isosceles and the circle becomes a line. When it is on the horizontal line, the triangle is degenerate and some of the four points don't exist. |