Prove that a Parallelogram Circumscribing a Circle is a Rhombus Last Updated : 23 Jul, 2025 Comments Improve Suggest changes 2 Likes Like Report Given: Parallelogram ABCDCircle PQRSTo prove: ABCD is a rhombus.Proof:We know that the tangents drawn from an exterior point to a circle are equal to each other. Therefore:AP = AS ⇢ (1)BP = BQ ⇢ (2)DS = DR ⇢ (3)CR = CQ ⇢ (4)Adding the LHS and RHS of equations 1, 2, 3, and 4:AP + BP + DS + CR = AS + BQ + DR + CQAB + DR + CR = AS + DS + BCAB + CD = AD + BCSince the opposite angles of a parallelogram are equal:2AB = 2BCAB = BC, and similarly, CD = AD.Therefore: AB = CD = BC = AD.Since all the sides are equal, ABCD is a rhombus.Read More:ParallelogramCircleTangent Create Quiz Comment S surajb1g23 Follow 2 Improve S surajb1g23 Follow 2 Improve Article Tags : Mathematics School Learning Maths Quadrilaterals Explore Basic ArithmeticWhat are Numbers? 15+ min read Arithmetic Operations 9 min read Fractions - Definition, Types and Examples 7 min read What are Decimals? 10 min read Exponents 8 min read Percentage 4 min read AlgebraVariable in Maths 5 min read Polynomials| Degree | Types | Properties and Examples 9 min read Coefficient 8 min read Algebraic Identities 14 min read Properties of Algebraic Operations 3 min read GeometryLines and Angles 9 min read Geometric Shapes in Maths 2 min read Area and Perimeter of Shapes | Formula and Examples 10 min read Surface Areas and Volumes 10 min read Points, Lines and Planes 12 min read Coordinate Axes and Coordinate Planes in 3D space 6 min read Trigonometry & Vector AlgebraTrigonometric Ratios 4 min read Trigonometric Equations | Definition, Examples & How to Solve 9 min read Trigonometric Identities 7 min read Trigonometric Functions 6 min read Inverse Trigonometric Functions 8 min read Inverse Trigonometric Identities 9 min read CalculusIntroduction to Differential Calculus 6 min read Limits in Calculus 11 min read Continuity of Functions 10 min read Differentiation 2 min read Differentiability of Functions 9 min read Integration 3 min read Probability and StatisticsBasic Concepts of Probability 6 min read Bayes' Theorem 13 min read Probability Distribution - Function, Formula, Table 11 min read Descriptive Statistic 5 min read What is Inferential Statistics? 5 min read Measures of Central Tendency in Statistics 9 min read Set Theory 3 min read PracticeNCERT Solutions for Class 8 to 12 7 min read RD Sharma Class 8 Solutions for Maths: Chapter Wise PDF 5 min read RD Sharma Class 9 Solutions 10 min read RD Sharma Class 10 Solutions 9 min read RD Sharma Class 11 Solutions for Maths 13 min read RD Sharma Class 12 Solutions for Maths 13 min read Like