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Proving the Parallelogram Circumscribing a Circle is a Rhombus

 

Introduction

A parallelogram is a quadrilateral with opposite sides parallel. A rhombus is a special type of parallelogram with all sides of equal length. In this article, we will explore the relationship between a circle and the parallelogram that circumscribes it. We will prove that this parallelogram is indeed a rhombus, providing a clear understanding of the geometric properties involved.

The Parallelogram Circumscribing a Circle

When a circle is inscribed within a parallelogram, the circle touches each side of the parallelogram at exactly one point. This is known as the circumscribing property. Let’s examine this relationship in more detail.

Property 1: Equal Opposite Angles

Consider a parallelogram ABCD with a circle inscribed within it. Let O be the center of the circle. Since opposite sides of a parallelogram are parallel, we can conclude that angle AOB is equal to angle COD, and angle BOC is equal to angle DOA. This is because the opposite sides of a parallelogram are parallel and the circle touches each side at exactly one point.

Property 2: Equal Opposite Sides

Let’s examine the lengths of the sides of the parallelogram. Since the circle touches each side of the parallelogram at exactly one point, the line segments from the center of the circle to the points of contact are perpendicular to the sides of the parallelogram. Let E, F, G, and H be the points of contact on sides AB, BC, CD, and DA, respectively.

By the properties of a circle, we know that the line segments OE, OF, OG, and OH are all radii of the circle and therefore have equal lengths. Additionally, since the line segments from the center of the circle to the points of contact are perpendicular to the sides of the parallelogram, we can conclude that line segments AE and CE are equal in length, as well as line segments BF and DF, CG and EG, and DH and AH.

Proving the Parallelogram is a Rhombus

Now that we have established the properties of the parallelogram circumscribing a circle, we can prove that it is indeed a rhombus.

Proof 1: Opposite Sides are Parallel

Since the parallelogram ABCD has opposite sides of equal length, we can conclude that AB is parallel to CD and BC is parallel to AD. This is a property of parallelograms.

Proof 2: All Sides are Equal in Length

By examining the lengths of the sides, we can see that AB = CD, BC = AD, and AB = BC = CD = AD. Therefore, all sides of the parallelogram are equal in length, which is a defining characteristic of a rhombus.

Proof 3: Diagonals Bisect Each Other at Right Angles

Let’s consider the diagonals of the parallelogram. The diagonals of a parallelogram bisect each other. Since the circle is inscribed within the parallelogram, the diagonals of the parallelogram pass through the center of the circle, which we denoted as point O.

By Property 1, we know that angle AOB is equal to angle COD, and angle BOC is equal to angle DOA. Since opposite angles of a parallelogram are equal, we can conclude that angle AOB is equal to angle BOC, and angle COD is equal to angle DOA.

Therefore, the diagonals of the parallelogram bisect each other at right angles, which is a property of a rhombus.

Conclusion

The parallelogram that circumscribes a circle is indeed a rhombus. We have proven this by examining the properties of the parallelogram and the circle inscribed within it. The equal opposite angles, equal opposite sides, and diagonals that bisect each other at right angles all contribute to the conclusion that the parallelogram is a rhombus.

Q&A

Q1: What is the definition of a parallelogram?

A1: A parallelogram is a quadrilateral with opposite sides parallel.

Q2: What is a rhombus?

A2: A rhombus is a special type of parallelogram with all sides of equal length.

Q3: How does a circle inscribe within a parallelogram?

A3: A circle inscribes within a parallelogram when the circle touches each side of the parallelogram at exactly one point.

Q4: What are the properties of the parallelogram circumscribing a circle?

A4: The properties of the parallelogram circumscribing a circle include equal opposite angles, equal opposite sides, and diagonals that bisect each other at right angles.

Q5: How can we prove that the parallelogram circumscribing a circle is a rhombus?

A5: We can prove that the parallelogram circumscribing a circle is a rhombus by demonstrating that it has opposite sides that are parallel, all sides of equal length, and diagonals that bisect each other at right angles.

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