Question 11

Parallelogram: A quadrilateral where opposite sides are parallel and equal, and opposite angles are equal.

Rhombus: A parallelogram where all four sides are equal. Every rhombus is a parallelogram, but not every parallelogram is a rhombus.

Circumscribing a circle: A polygon circumscribes a circle when the circle fits exactly inside it, touching all sides. Each side of the polygon is a tangent to the circle.

Step 1 — Identify properties of the parallelogram

Let $ABCD$ be a parallelogram circumscribing a circle. This means all its sides touch the circle. Let the points of contact be $P$, $Q$, $R$, and $S$:

• Point $P$ is on side $AB$.
• Point $Q$ is on side $BC$.
• Point $R$ is on side $CD$.
• Point $S$ is on side $DA$.

In a parallelogram, opposite sides are equal. Therefore: $AB = CD \quad \text{(Equation 1)}$ $BC = AD \quad \text{(Equation 2)}$

Step 2 — Apply tangent properties

Tangents from an External Point are Equal: Any two tangents drawn from the same external point to a circle are always equal in length. Each vertex of ABCD is an external point with two tangents to the circle.

We know that the lengths of tangents drawn from an external point to a circle are equal.

From vertex $A$: $AP = AS \quad \text{(Equation 3)}$

From vertex $B$: $BP = BQ \quad \text{(Equation 4)}$

From vertex $C$: $CR = CQ \quad \text{(Equation 5)}$

From vertex $D$: $DR = DS \quad \text{(Equation 6)}$

Step 3 — Sum the tangent lengths

Adding Equations 3, 4, 5, and 6: $AP + BP + CR + DR = AS + BQ + CQ + DS$

Grouping the terms on both sides: $(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)$

From the figure, we can see that:

• $AP + BP = AB$
• $CR + DR = CD$
• $AS + DS = AD$
• $BQ + CQ = BC$

Substituting these values into the equation: $AB + CD = AD + BC$

Using Equation 1 ($CD = AB$) and Equation 2 ($AD = BC$): $AB + AB = BC + BC$

$2AB = 2BC$

Dividing both sides by $2$:

$AB = BC \quad \text{(Equation 7)}$

Step 4 — Conclude the shape

From Equation 1, we have $AB = CD$. From Equation 2, we have $BC = AD$. From Equation 7, we have $AB = BC$.

Combining these equations: $AB = BC = CD = DA$

Since all four sides of the parallelogram $ABCD$ are equal, it is a rhombus.

Answer

The parallelogram circumscribing a circle has all its sides equal. Hence, it is a rhombus.