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Construction on Triangle
Posted:
Dec 30, 1994 8:36 PM


There are many constructions on triangles which result in equilateral triangles, but here's one that generalizes nicely. On each side of a triangle, construct outwardpointing isoceles triangles with aps angle 120 degrees with their base (unequal side) on the original triangle. The apices of the three new triangles form an equilateral triangle. I think this is quite hard to prove by usual plane geometry techniques, but there's an elegant answer. (This is by no means a new or original problem.) Now generalize this to a quadrilateral. On each of its sides construct isoceles triangle of apex angle 90 degrees (360/4) with their bases on the sides of the original quadrilateral. This forms a new quadrilateral (NOT a square yet). On the sides of this construct isoceles traingle of apex angles 180 (in other words, bisect its sides). THIS is a square. Generalization to 5 sides: First construction uses apex angle of 360/5; second construction (on the sides of the first) uses apex angles 2*360/5. Third and last uses apex angles of 3*360/5 (these actually point inwards). This last pentagon is regular. This holds true in a similar way for N2 constructions starting with an arbitrary Ngon. You finally get a regular Ngon. This has, believe it or not, an 'elementary' proof (I didn't say easy!). My proof is unfortunately algebraic. Not quite the FLT, but one of the few geometry problems I know of that has a free integral parameter of this sort. Have fun with this (at your peril)or ignore it.  Steve Gray Graphics and Geometry Consultant Santa Monica CA



