It is interesting to note that the diameters of the circle are the angle bisectors of the three angles of the triangle in the following diagram.
The center of the circle* is also the center of the triangle**.
Knowing this, given any equilateral triangle one finds its center by bisecting the three angles.*** The distance from the center of the triangle to a vertex will also be the radius of the circle that will inscribe the triangle.
Looked at in this way, one can see the affinity the equilateral triangle has to both the hexagon and the circle. It would almost seem that this triangle is the hexagon reduced to its lowest common denominator, a distilled hexagon .
*The center of a circle in which a polygon is inscribed is called the circumcenter.
**The point where the three angle bisectors of a triangle meet is called the incenter. In an equilateral triangle, the incenter will coincide with both the circumcenter of a circle in which the triangle is inscribed, and with the center of a circle inscribed within it.
***The circumcenter of triangles other than equilateral triangles coincides with the point of intersection of the perpendicular bisectors of the sides.
If one looks at the inscribed square in relation to the inscribed octagon, one can find parallels.
The diagonals of the octagon that are not diameters outline a set of two squares; however, the simplest way to construct an inscribed octagon is to make an inscribed square first, not the other way around. What does this reveal?
What this suggests is that there is an ordinal sequence to the making of geometric constructions of inscribed regular polygons. The hexagon seems to be the most integral to the anatomy of the circle. The triangle we derive is the most primal of all odd-sided inscribed polygons, and the square is the most primal of even-sided inscribed polygons. Although the hexagon is necessary to the construction of the inscribed triangle, the octagon is not necessary to the construction of the inscribed square.
Inscribed triangles: Sketch (33K)
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