Question

prove that the circle drawn with any side of a rhombous as diameter passes through the point of intersection of its diagonal

Answer 1

We have to prove that circle drawn on diameter AB will pass through O. => A circle drawn with O as center and radius OApasses through A, Q, B. So the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.

Let ABCD is a rhombus having two diagonals AC and BD which bisect each other at right angles.

We have to prove that circle drawn on diameter AB will pass through O.

Now from Q, draw PO || Ad and EF || AB

Again from figure,

AB = CD

=> AB/2 = CD/2

=> AO = DP  (since O and P are the mid points of AB and CD)

Similarly  AE = OQ

=> AO = OQ = OB

=> A circle drawn with O as center and radius OA passes through A, Q, B.