A circle touches all sides of a parallelogram. So the parallelogram must be a, ______.

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A circle touches all sides of a parallelogram. So the parallelogram must be a, rhombus. Explanation: ABCD is a parallelogram. A circle with centre O touches the parallelogram at E, F, G and H. ∴ AB = CD ...(1) (Opposite sides of parallelogram are equal) AD = BC ...(2) (Opposite sides of parallelogram are equal) Tangent segments drawn from an external point to a circle are congruent. AE = AH ...(3) DG = DH ...(4) BE = BF ...(5) CG = CF ...(6) Adding (3), (4), (5) and (6), we get AE + BE + CG + DG = AH + DH + BF + CF ⇒ AB + CD = AD + BC ...(7) From (1), (2) and (7), we have 2AB = 2BC ⇒ AB = BC ...(8) From (1), (2) and (8), we have AB = BC = CD = AD ∴ Parallelogram ABCD is a rhombus. ...(A rhombus is a parallelogram with all sides equal.) Hence, the correct answer is rhombus.

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