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