Math CentralQuandaries & Queries


Question from Yueh, a student:

Let ABC be a triangle. Locate the point O inside the triangle ABC such that for points L, M and N lying on its sides AB, BC, and CA respectively, triangles BOL, COM, and AON have the same area, where LO and BC, OM and AC, NO and AB must be parallel respectively.


The first step is to GUESS where O has to be for the triangles BOL and AON to have the same area. I would guess that O must lie on the line joining C to the midpoint of AB. (Try to prove whether my guess is correct by extending the line ON to the point N' where it meets BC. Then LBN'O is a parallelogram. What can you say about the areas of triangles BOL and BON'? What about BON' and AON?)


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