Aim: To construct 1/3 ( also 1/n)
Let x=1/3 and assume that you know how to bisect a line segment.
then 3x=1 ==> -3x= -1
adding x^2+y^2 on both sides.
===>
x^2+y^2 - 3x= x^2+y^2 - 1
==> (x-3/2)^2+y^2-(3/2)^2 = x^2+y^2 - 1
==> S1=S2
i.e. Circle S1 of radius 3/2 centered at (3/2, 0) = unit circle S2 at origin
Hence the line 3x-1=0 will be the common chord of S1 and S2.
So now draw those two circles S1 and S2 using Ruler-Compass to find the intersecting points P1 and P2.
Join P1 and P2 that will cut the X-axis at ordinate x=1/3.
QED
Post Script (P.S.) : Same can be generalized for x=1/n for any integer n
Let x=1/3 and assume that you know how to bisect a line segment.
then 3x=1 ==> -3x= -1
adding x^2+y^2 on both sides.
===>
x^2+y^2 - 3x= x^2+y^2 - 1
==> (x-3/2)^2+y^2-(3/2)^2 = x^2+y^2 - 1
==> S1=S2
i.e. Circle S1 of radius 3/2 centered at (3/2, 0) = unit circle S2 at origin
Hence the line 3x-1=0 will be the common chord of S1 and S2.
So now draw those two circles S1 and S2 using Ruler-Compass to find the intersecting points P1 and P2.
Join P1 and P2 that will cut the X-axis at ordinate x=1/3.
QED
Post Script (P.S.) : Same can be generalized for x=1/n for any integer n
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