A Reuleaux triangle is a shape formed from the intersection of three circles, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"
Beginning with the above equilateral triangle (gray) with sides 300px, we use this length as the radius of three arcs, centered on each corner. These are the blue arcs, which incidentally is the Reuleaux triangle.
Being a curve of constant width means the Reuleaux will rotate inside a square (green), WHILE TOUCHING ALL 4 SIDES. However, the center of the rotation is not a single point. It floats, and is composed of parts from four ellipses.
The Reuleuax center must follow this path (shown in black) as it rotates. This is how to drill a square hole - the chuck 'floats' in this path.
This is indicated by the black squaroid, which is actually composed of 4 elliptical arcs.
Interesting mechanical devices have used this feature, such as drilling square holes, a combustion engine, and a film projector.
We will examine this further.
Some formulae:
Centroid, circumcircle center, incircle center, medians, perpendicular bisectors, angle bisectors & heights all coincide at 1 point (red lines).
This center is 1/3 of the height, measured from any side.
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