The fact that a vertex angle of a triangle can be divided (bisected, trisected, etc.) in a specified manner before the opposite side is finally fixed (imagining the triangle to be under gradual construction), is consistent with the apparent situation that a result based on vertex-angle division is ‘stronger’ than the corresponding result for division of the opposite side. For example, the three vertex-angle bisectors meet at the incenter of the triangle, whereas the three side bisectors (medians) meet at the centroid (balance point). Arguably, the center of the inscribed circle is ‘more important’ than the balance point. Again, the vertex-angle trisectors give rise to an equilateral triangle, whereas the side trisectors give rise to a similar triangle of 1/5 the linear length of the original triangle. Here, there is even less doubt about the relative importance: the former is known as ‘Morley’s miracle’, whereas the latter seems merely awesome.

keywords: Logic, Mathematics, Geometry

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