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Circles

Short Answer Type

61. Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see figure). Prove that ∠ACP = ∠QCD.

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62. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side.

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63.

ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.

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64.
Prove that a cyclic parallelogram is a rectangle.

Given: ABCD is a cyclic parallelogram.
To Prove: ABCD is a rectangle.

Proof: ∵ ABCD is a cyclic quadrilateral
∴ ∠1 + ∠2 = 180° ...(1)
| ∵ Opposite angles of a cyclic quadrilateral are supplementary
∴ ABCD is a parallelogram
∴ ∠1 = ∠2 ...(2)
| Opp. angles of a || gm
From (1) and (2),
∠1 = ∠2 = 90°
∴ || gm ABCD is a rectangle.

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