Let us prove that if the circles are drawn having sides of a rhombus as diameter then the circles pass through the fixed points.
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Let ABCD be a rhombus whose diagonals AC and BD intersect at O.
We know that, diagonals of a rhombus intersect each other at right angle.
∴ ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°
Now, circles with AB, BC, CD and DA as their respective diameters pass through O. (Angle in a semi-circle is 90°)
Thus, the circles described on the four sides of a rhombus as their respective diameters, pass through the point of intersection of its diagonals.
Hence, proved
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