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Topic: Construction on Triangle
Replies: 1   Last Post: Jan 5, 1995 12:45 AM

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steve gray

Posts: 17
Registered: 12/6/04
Construction on Triangle
Posted: Dec 30, 1994 8:36 PM
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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 outward-pointing
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 N-2 constructions
starting with an arbitrary N-gon. You finally get a regular
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

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