Proof that the Rhombus inscribed in a circle is a square:

To prove that a rhombus inscribed in a circle is a square, we need to show that all four sides of the rhombus are congruent and all angles are right angles.

Step 1: Inscribed Angle Property
- Start by considering the properties of an inscribed angle. In a circle, an inscribed angle is equal to half the measure of the intercepted arc.
- Since a rhombus inscribed in a circle has opposite angles that are congruent, each angle of the rhombus is half the measure of the intercepted arc.
- Therefore, all four angles of the rhombus are congruent.

Step 2: Opposite Angles of a Rhombus
- In a rhombus, opposite angles are congruent.
- Since all four angles of the rhombus are congruent, it implies that opposite angles are right angles.

Step 3: Diagonals of a Rhombus
- The diagonals of a rhombus are perpendicular bisectors of each other.
- Since opposite angles of the rhombus are right angles, the diagonals are perpendicular to each other.

Step 4: Congruent Sides
- The diagonals of a rhombus also bisect each other, forming four congruent right triangles.
- Each right triangle has two congruent sides (the diagonals) and the hypotenuse (side of the rhombus).
- By the hypotenuse-leg congruence theorem, the four sides of the rhombus are congruent.

Conclusion:
- We have shown that all four angles of the rhombus are right angles and all four sides are congruent.
- By definition, a quadrilateral with four right angles and congruent sides is a square.
- Therefore, a rhombus inscribed in a circle is a square.