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The diagonals of a cyclic quadrilateral ...

The diagonals of a cyclic quadrilateral are at right angles. Prove tha the perpendicular from the point of their intersection on any side when produced backwards , bisects the opposite side.

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Let ABCD be a cyclic quadrilateral whose diagonals AC and BD intersect at O at right angles . Let `OL _|_ AB` such that LO produced meets CD at M.
Then, we have to prove that CM `=` MD
Clearly, `/_ 1 = /_ 2 [ /_ s` in the same segment]
`/_ 2 + /_ 3= 90^(@) [ :' /_ OLB = 90^(@) ]`
`/_ 3 + /_ 4 = 90^(@) [ :' `LOM is a straight line and `/_ BOC = 90^(@) ]`
Thus, `/_ 1 = /_ 2` and `/_ 2 = /_4 implies /_1 = /_ 4`
`:. `OM `=` CM and similarly, `OM = MD`.
Hence `CM = MD`
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