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    Re: elementary geometry question
    From: Chuck Taylor
    Date: 1995 Aug 1, 10:00 -0700

    Josiah Slack (slack@stonehand.com) wrote:
    > Last summer, a friend let me peruse a book of his called "Shoreline and
    > Sextant".  One of the useful techniques therein involved finding three marks, A,
    > B and C, measuring the angles AB and BC, constructing "circles of equal angle"
    > on the chart, and noting the places where the circles intersected - one of which
    > represents your position.  Bowditch describes the overall procedure, but with a
    > different method of constructing the circles.  The method described in
    > "Shoreline and Sextant" seemed simpler and more elegant to me, but I've gone and
    > forgotten it.  And it's been a long time since I've done
    > straightedge-and-compass work.  Does anyone know the construction I'm talking
    > about?
    > Thanks in advance.
    > Josiah Slack (slack@stonehand.com)
    The following method is from the Harvard Book ("Learn to Navigate by the
    Tutorial System Developed at Harvard" by Whitney & Wright).
    Suppose your 3 points are A, B, & C and the angle between A & B is a1
    and the angle between B & C is a2.
    Draw a straight line between A & B. With a protractor and straightedge,
    construct a line from point A at an angle (90 - a1) to the line between
    A & B, and another line at B, also at an angle (90 - a1) to AB.
    Construct these lines so that they intersect, as in the following;
                 \        /
                  \      /
                   \    /
                    \  /
    Draw the lines so that O1 is generally toward you, as opposed to away
    from you (on the other side of A & B).
    The point where these two lines intersect (O1) is the center of the
    first circle of position. Set your compass to the radius O1-A or O1-B
    (they will be the same) and draw the circle of position.  Repeat the
    process with B, C, & a2 to find O2, and draw the second circle of
    position. The two circles will intersect at two points, one of which is
    your position and the other of which is one of the 3 points (A, B, or C).
    This method fails if O1 and O2 happen to coincide, in which case all you
    have is one circle of position, not a fix.
    This method works because geometry tells us that a circle can be
    constructed through any 3 points (so long as they don't all lie on a
    straight line). Two of the points in the case of the first circle of
    postion are A & B and the third is your position. In the case of the
    second circle of position, the first two points are B & C and the third
    is your position.
    This method can also be used with two pairs of points, AB and CD. In
    that case the two circles will intersect at two possible fixes, but it
    is usually obvious which one to choose.
    There is another method that I haven't seen in print anywhere (I wrote
    it up and sent it to Ocean Navigator, but they didn't see fit to print
    it). It involves a little trig and a pocket calculator. Assume the same
    same setup as above.
    1. Measure the distance AB on your chart.
    2. Measure the angle a1 between A & B.
    3. Compute r1 = AB/(2*sin(a1)).
    4. Set your compass to radius r1 using the distance scale on your chart.
    5. Draw an arc of radius r1 centered at A and another centered at B.
    Where these two arcs intersect is O1, the center of the first circle of
    position. Without changing the setting of the compass, move the point to
    O1 and draw the first circle of position.
    6. Repeat for B, C, and a2.
    Chuck Taylor
    Everett, WA

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