Astroid is Envelope of Line Segment whose End Points lie on Coordinate Axes

Theorem

Let $\LL$ be a line segment of length $a$.

Let the endpoints of $\LL$ be constrained to lie on the coordinate axes of a Cartesian plane: one on the $x$-axis and one on the $y$-axis

The envelope of $\LL$ forms an astroid whose cusps lie on the points at which the coordinate axes intersect a circle of radius $a$ whose center is at the origin.

Proof

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Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): astroid