In the adjoining figure, a point O is taken inside an equilateral quad. ABCD such that OB = OD. Show that A, O and C are in the same straight line.

Explanation

Given A quad. ABCD in which AB = BC = CD = DA and O is a point within it such that OB = OD.
TO PROVE `angle AOB + angle COB = 180^(@)`.
PROOF In `triangle OAB and OAD,` we have
AB = AD (given ), OA = OA (common) and OB = OD (given)
`therefore triangle OAB ~= triangle OAD. `
`therefore angle AOB = angle AOD." " ` ...(i)(c.p.c.t.)
Similarly, `triangle OBC ~= triangle ODC. `
`therefore angle COB = angle COD. " " `...(ii)
Now, `angle AOB +angle COB +angle COD + angle AOD=360^(@) " " [angle " at a point"]`
`rArr 2 (angle AOB +angle COB)=360^(@) rArr angle AOB +angle COB=180^(@).`