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A point O is taken inside an equilateral...

A point `O` is taken inside an equilateral four sided figure `A B C D` such that its distances from the angular points `D` and `B` are equal. Show that `A O` and `O C` are in one and the same straight line. GIVEN : A point `O` inside an equilateral quadrilateral four sided figure `A B C D` such that `B O=O Ddot` TO PROVE : `A O` and `O C` are in one and the same straight line.

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Given: A point `O` inside an equilateral quadrilateral four sided figure `ABCD` such that `BO=OD`
To prove: `AO` and `OC` are in one and the same straight line.
Proof:
In `/_\AOD` and `/_\AOB`,
`AD=AB` (Given)
`AO=AO` (Common side)
`OD=OB` (Given)
`/_\AOD=~/_\AOB` (`SSS` congruence rule)
`/_AOD=/_AOB` (`CPCT`)
Similarly, `/_\DOC=~/_\BOC` (`SSS` congruence rule)
`/_DOC=/_BOC` (`CPCT`)
`/_AOD+/_AOB+/_DOC+/_BOC=360^@` (Angles at a point)
`2/_AOD+2/_DOC=360^@`
`/_AOD+/_DOC=180^@`
Hence, `AO` and `OC` are in one and the same straight line.
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