Experiment: Dividing a segment with a ruler alone

In the figure below, begin with collinear A, B, C.

• Choose any O not on the line.
• Draw any line through C not through O (or A and B).  Let the intersections be A' and B'.
• Draw AB' and BA' and let P be the intersection.
• Draw OP and let D and C' be the intersections in the figure.

Do this again with an entirely different O and CA' and see where the new D ends up.

Ratio CA/CB and the Law of Sines

In the figure below, apply the Law of Sines to triangles OCA and OCB to get a relationship among the quantities CA, CB, OA, OB, and the sines of some angles. Use this to write CA/CB in terms of OA, OB and some sines.

Ratio DA/DB and the Law of Sines

In the figure below, apply the Law of Sines to triangles ODA and ODB to get a relationship among the quantities DA, DB, OA, OB, and the sines of some angles. Use this to write DA/DB in terms of OA, OB and some sines.

• Look carefully at the + or – sign in the ratio of lengths to get a correct formula.

Ratio (CA/CB)/(DA/DB) and the Law of Sines

In the figure below, combine both results from before to get a formula for (CA/CB)/(DA/DB).  See what is left when you cancel everything.

• If CD divides AB harmonically, what is (CA/CB)/(DA/DB)? (Use the definition!)

Main Point:  Find the ratio (C'A'/C'B')/(D'A'/D'B') without doing any additional work

Think through what the formula will be for this ratio and compare it with the other formula. 

• If (CA/CB)/(DA/DB) = 3, what is  (C'A'/C'B')/(D'A'/D'B')?