Draw straight line OM

Draw circle centred at O and radius OM

Draw another circle centred at M with radius OM

P and Q are intersection points of the two circles

    OP = OM          radius

    MP = OM          radius

So  MP = OM = OP

So  ∇OPM is equilateral

Similarly ∇OQM is equilateral 

And ∇OPM is isoscelese   as MP = MO

So  ∠MPO = ∠MOP 

And ∇OPM is isoscelese   as MP = PO

So  ∠PMO = ∠MOP

So  ∠PMO = ∠MOP = ∠MPO = 60º  sum ang. tri. = 180º

Similarly 

    ∠QMO = ∠MOQ = ∠MQO = 60º

Draw PQ which intersects MO at N

    ∇OPQ is isoscelese     as OP = OQ

So  ∠OPQ = ∠OQP

    ∠POM = ∠QOM = 60º      proven above

      NO = NO

so  ∇OPN = ∇OQN            2 sides & inc. ang. eq.

So  ∠ONP = ∠ONQ = 90º      ang. of str. line = 180º

    ∠NPO = ∠NQO = 30º      sum ang. tri. = 180º

Similarly

    ∇MPN = ∇MQN

    ∠NPM = ∠NQM = 30º

Let PO = 1

   MN = ON = ½           proven previously

   PN² = PO² - NO²       Pythagoras

   PN² = 1² - ½²

       = 1 - ¼

       = ¾

So

    PN = (√3)/2