Question
Using vectors, prove that the perpendicular bisectors of the sides of a triangle are concurrent.
Solution
Let ABC be the given triangle and O be the point of intersection of perpendicular bisectors OD and OE of sides BC and CA respectively.
Let F be mid-point of AB. Join O to F.
Take O as origin.
Let 
be the position vectors of A, B, C respectively.
The position vectors of D, E, F are 
respectively.
Since 
Again, 
Adding (1) and (2), we get,
perpendicular bisectors meet in a point.
