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    Re: Finding The Symmedian
    From: Herbert Prinz
    Date: 2010 Dec 18, 20:06 +0000

    On 2010-12-17 18:22, Greg Rudzinski wrote:
    >
    > The linked diagram shows an alternate route to finding the symmedian
    > point by using two sides with corresponding vertex. A short spread on
    > a divider compass expedites bisecting of angles without the aid of a
    > protractor. Dividing a side in half is done by a few trials of the
    > divider. Transposing the bisected vertex and bisected side is
    > facilitated with a circle.
    >
    > It is not practical to be using a symmedian point when laying down a
    > round of stars but I do see a place for it where a round of GPS
    > satellites are automatically analyzed for ultra precise positioning.
    >
    >
    
    I find it hard to believe that there be little Gremlins in our GPS
    receivers plotting the solution to the minimum square sum problem. Most
    GPS receivers  nowadays use algebraic solutions. Plotting is meant for
    people, especially for those of us who would rather approximate a
    solution within the accuracy limits of a rough plot than tackle an
    involved algebraic solution.
    
    In some of the recent posts I have come across a strange notion that
    "eyeballing" is a method in its own right to determine the position of
    the MPP,  quasi an alternative to various constructive methods. How can
    this be? How can one find a point by "eyeballing", when one does not
    know what to look for in the first place? Once you know that the point
    in question is the symmedian point , you can eyeball it easily by
    exploiting some of its unique properties. Or by doing the construction
    in your head. Experience helps. "Eyeballing" is a way of implementing a
    given method, not an alternative to it.
    
    Several more or less obscure alternatives to the symmedian point have
    been proposed without any mathematical foundation or reasoning
    whatsoever. It is totally backwards to come up with a substitute for the
    symmedian point that is arguably more or less different from the MPP
    (dependent on the geometry of the situation) and then plot this wrong
    point with perfect accuracy. It makes by far more sense to find a quick
    and dirty method to plot the correct point.
    
    Anybody who takes the time to look at Villarceau's diagram that Frank
    has kindly posted will be hard pressed to come up with an easier or
    faster construction. You don't even need dividers! As I mentioned before
    here on this forum, his construction is based on the property of the
    symmedian point that x:y:z = a:b:c, where x,y,z are its distances from
    the sides a,b,c.
    
    The only tool you need for Villarceau's method is a  90-45-45
    plexiglass  plotting triangle. It has with a scale on its hypotenuse
    running outwards in both directions from the center, and it has a
    perpendicular grid for drawing the parallels and perpendicular lines.
    Leave the dividers to Euclides and the purists! To cut down on the
    required plotting area, I recommend:  1) Instead of plotting the
    parallels at distances a,b,c, use the half distances. That's where the
    symmetric scale of the plotting triangle comes in handy. 2) Instead of
    plotting the parallels outside of the triangle, plot them on the other side.
    
    Not practical? You must be kidding.
    
    Coming back on "eyeballing". When I spoke in Mystic about finding the
    MPP geometrically, I started out with drawing a rectangle on the
    blackboard. Then I passed a laser pointer around the audience and
    everyone had to "eyeball" the MPP. I marked every guessed point with
    chalk. You can see the results at
    
    http://www.fer3.com/mystic2008/afterphotos.html
    
    Scroll down until you see me scribbling on a blackboard with some
    drawings and then zoom in on the triangle in the top left corner. The
    correct point is at the cross. Most people tend to be too far near the
    acute corner. One person guessed it right. It should not come as a
    surprise that he is an experienced navigator. You will also see that I
    cheated: How could I be sure that I would find the MPP instantly with
    perfect accuracy? My triangle has a right angle. At the end of the
    presentation, I drew the altitude to the hypotenuse, cut it in half, and
    voila!
    
    Every triangle that I have ever seen is close to one of the following
    special cases: Equal sides, isosceles, rectangular, degenerate (i.e.
    very very acute, very very obtuse). Know where the symmedian point is in
    each of these trivial cases and you will never fail to make an educated
    guess about the MPP. Art or science? In the olden days there was no
    difference.
    
    Herbert Prinz
    
    
    
    

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