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Re: Bisecting lines
Posted:
Dec 30, 2001 3:44 AM
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On Sat, 29 Dec 2001, John Conway posed the following problem:
> The lines that do exactly bisect the triangle envelop a
> little deltoid-shaped region formed from segments of three
> hyperbolas, the vertices being the three midpoints of the
> medians (or equivalently, they are the vertices of the
> medial triangle of the medial triangle). Since the figure
> is affinely invariant, the area of this region is a fixed
> proportion of the area of the whole triangle - perhaps
> someone will care to work out just what this proportion is?
In a followup message he then described the solution:
> I did so, finding the answer to be
>
> (3.log2 - 2)/4 = .019860...
>
> if I wasn't mistaken. The calculation was much easier
> than I expected, and I'll repeat it here as a check,
> since that area (under 2%) seems suspiciously small.
John, the (3 ln 2 - 2)/4 appears to be correct; I made an
independent calculation and arrived at the same answer.
I worked exclusively with an equilateral triangle and
ploughed through the calculation with brut force rather
than using your clever trick with the right triangle.
The 2% answer indeed appears to be surprisingly small,
so for the purpose of visual inspection I made a plot
of an equilateral triangle and the corresponding deltoid.
It can be found in:
http://www.math.umbc.edu/~rouben/deltoid.html
The blue triangle in the lower-left corner has an area
which is exactly 2% of the area of the big triangle.
It seems to be commensurate with the deltoid in the
eyeball metric.
--
Rouben Rostamian <rostamian@umbc.edu>
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