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Re: Euler's Line
Posted:
Dec 29, 2001 3:08 PM
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On Mon, 24 Dec 2001, Ben Saucer wrote:
> At 07:05 AM 12/08/2001, you wrote:
> >Within any given triangle: angle bisectors, medians, and perpendicular
> >bisectors each cross at their own respective points. Each of these points
> >falls upon a line: Euler's Line.
> >When I was in tenth grade(1984-85), I postulated that Euler's Line divided
> >any given triangle into two equal areas. Was I wrong?
>
> That's probably true of ANY line which passes through the median center.
... except that it isn't! I sent out the true statement some
time ago, and will now amplify it. A line through G bisects the
triangle into two parts with equal area if and only if it is one of
the three medians. Otherwise, it cuts it into a triangular part
whose area is between 4/9 and 1/2 of the whole, and a quadrilateral
part whose area is between 1/2 and 5/9 of it, the extreme cases
(4/9 : 5/9) arising just when the line is parallel to a side.
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?
John Conway
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