Q21 of 100 Page 17

Prove that the circles described on the four sides of a rhombus as diameters, pass through the point of intersection of its diagonals.

Let ABCD be a rhombus such that its diagonals AC and BD intersects at O

Since, the diagonals of a rhombus intersect at right angle


Therefore,


∠ACB = ∠BOC = ∠COD = ∠DOA = 90o


Now,


∠AOB = 90o = circle described on AB as diameter will pass through O


Similarly, all the circles described on BC, AD and CD as diameter pass through O.


More from this chapter

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19

Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.

 20

Prove that the centre of the circle circumscribing the cyclic rectangle ABCD is the point of intersection of its diagonals.

 22

If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are equal.

 23

ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA=ED. Prove that:

(i) AD||BC (ii) EB=EC