Sat March 07, 2015 By: Sujay Sarkar

Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point intersection of its diagonals.

Expert Reply
Mili Hariyani
Sat March 07, 2015

Let PQRS be a rhombus with diagonals PR and QS which intersect at O.

We know that diagonals of rhombus intersect each other at right angles. Therefore

begin mathsize 12px style angle P O Q space equals space angle Q O R space equals space angle R O S space equals space angle S O P space equals space 90 degree end style

Here we have circles passing through PQ, QR, RS, SP as diametre and they pass through O.

For the circle with PQ as diameter, we have angle POQ inscribed in the semi-circle. Also angle inscribed in a semi-circle is a right angle.

Thus, the circle passes through O.

This is true for any circle constructed on any of the side of rhombus.

Therefore, circle with the side of rhombus as diameter passes through the point of intersection of its diagonals.

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