Parbelos

The parbelos is a figure similar to the arbelos but instead of three half circles it uses three parabola segments.

More precisely the parbelos consists of three parabola segments, that have a height that is one fourth of the width at their bases.

of the three parabola arcs is a parallelogram the area of which relates to the area of the parbelos as follows:[1] The four tangents at the three cusps of the parabola intersect in four points, which form a rectangle being called the tangent rectangle.

The circumcircle of the tangent rectangle intersects the base side of the outer parabola segment in its midpoint, which is the focus of the outer parabola.

One diagonal of the tangent rectangle lies on a tangent to the outer parabola and its common point with it is identical to its point of intersection with perpendicular to the base at the inner cusp.