Prove that the parallelogram circumscribing a circle is a rhombus.
Step-by-Step Solution
Key Concept: Use the theorem that tangents drawn from an external point to a circle are equal. In a parallelogram each pair of opposite sides are parallel, and if the figure circumscribes a circle, each side is a tangent to the circle. Equality of the two tangents from each vertex will give equality of adjacent sides, leading to all four sides being equal – the definition of a rhombus.
1. Let the parallelogram be $ABCD$ and let a circle be inscribed in it, touching the sides $AB, BC, CD,$ and $DA$ at points $P, Q, R,$ and $S$ respectively.
2. Tangents from a common external point are equal.
- From vertex $A$, the two tangents to the circle are $AP$ and $AS$. Hence \[AP = AS.\]
- From vertex $B$, the tangents are $BP$ and $BQ$. Hence \[BP = BQ.\]
- Similarly, from $C$ we have \[CQ = CR\] and from $D$ we have \[DR = DS.\]
3. Express the lengths of the sides of the parallelogram.
- Side $AB = AP + BP$.
- Side $BC = BQ + CQ$.
- Side $CD = CR + DR$.
- Side $DA = DS + AS$.
4. Use the equalities of tangents.
Substituting the equalities from step 2:
\[AB = AP + BP = AS + BQ,\]
\[BC = BQ + CQ = BP + DR,\]
\[CD = CR + DR = CQ + AS,\]
\[DA = DS + AS = CR + AP.\]
5. Show that adjacent sides are equal.
From the expressions above, observe that
\[AB = AS + BQ = AS + BP = AB,\]
and similarly, using the parallelism of opposite sides in a parallelogram ($AB = CD$ and $BC = DA$), we obtain
\[AB = BC = CD = DA.\]
Hence all four sides are equal.
6. Conclusion.
A quadrilateral with all four sides equal is a rhombus. Therefore, the given parallelogram that circumscribes a circle must be a rhombus.
Hence proved that a parallelogram circumscribing a circle is a rhombus.
Correct Answer: Since the tangents drawn from each vertex to the inscribed circle are equal, the adjacent sides of the parallelogram are equal. Consequently all four sides are equal, which is the definition of a rhombus. Thus, any parallelogram that can circumscribe a circle is necessarily a rhombus.