Inspired by Sharky Kesa
Geometry
Level
4
How many steps are required to divide a line segment in the ratio with the help of a collapsible compass and a straightedge?
For the definitions see Sharky's Note .
12
7
10
8
14
11
9
13
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1 solution
Let the line segment be AB.
Through A draw a line XY passing through A.
With A as center and any arbitrary radius draw a circle to intersect the line XY at the points O and P.
With P as center and PO as radius draw a circle to intersect XY at Q.
Similarly with Q as center and QP as radius draw a circle to intersect XY at R.
And with R as centre and RQ as radius draw a circle to intersect XY at S.
Join SB.
Taking S as center and SR as radius draw a circle to intersect SB at T.
With T as centre and TS as radius draw an another circle to intersect the circle formed in step 5 at U.
Join RU to intersect AB at D.
.'. AD : DB = 5 : 2
The way shown above is a classical method of dividing a line segment in a ratio with the help of a compass and a ruler. I have extended it to collapsible compass and stretedge by adding circles. I have reduced the moves as far as possible. As it is a construction question and not any ordinary geometry problem, it can't be proved that the solution is shortest. I think it is the shortest possible solution and a shorter solution is IMPOSSIBLE .