Math CentralQuandaries & Queries


Question from Carolyn, a student:

I am doing construction. I have a line segment drawn. I need to divide a segment into 3 congruent part. How do I do that?


The key is in similarities, in particular similar triangles.


Start with a triangle ABC and a point D on BC so that the ratio of the lengths of BD to DC is p to q, in your case take p = 1 and q = 2. Construct a line through D parallel to CA and meeting AB at E. Construct a line through D parallel to AB and meeting CA at F. AEDF is a parallelogram and hence AE and DF are congruent.

The triangles CFD and DEB are similar (do you see why?) and hence

|EB|/|FD| = |BD|/|DC| = p/q

Does this help?

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