What is the only parallelogram that can be inscribed in a circle?

Answer 1

Any parallelogram can be inscribed in a circle if the parallelogram is sufficiently small, but only two of the "corners" (a corner is a vertex) of the parallelogram will lie on the circle. But any parallelogram with four right angles (a rectangle or a square) can be inscribed in a circle, and all four of the vertexes will lie on the circumference. So the only parallelogram that can be inscribed in a circle is a rectangle.

You'll recall that a parallelogram is a quadrilateral with two pairs of parallel sides. If the interior angles of a parallelogram are right angles, that sets conditions for a special case of a parallelogram called a rectangle. If the sides of a given rectangle are the same length, that rectangle is now a special case of rectangle called a square. Any rectangle (including the special case of the square) can be inscribed inside a circle so all vertexes lie on the circle.

If we're interested in a construction project, start by drawing a circle. Then pick any two points on the circle and connect them with a line segment. Next, draw a line segment from each of the original points across the circle, insuring that each line segment is at a right angle to that first line segment. Lastly, connect the two points on the circle where those last two line segments have interesected the circle. You'll find that in every case you try, you'll have constructed a rectangle. And if the line segments all end up the same length, your rectangle will be a square.

Answer 2

Rhombus