Given three points whose coordinates are known, and a point O inside the triangle they form, how do you...

geometry/construction/interior.point New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Given three points whose coordinates are known, and a point O inside the
triangle they form, how do you find the location of O if you know only
the angles at O subtended by each pair of the three points?

                               X (point 1)
                               |
                               |
                               |
                               O  <------ This angle (and the others like it)
                              / \         are known
                             /   \
                            /     \
                           /       \
                 (Point 2)X         \
                                     \
                                      X (Point 3)

[ Editor: problem statement reworded from the 1985 net.puzzle posting
"Any surveyors out there???" by Ross M. Greenberg
<greenber@timeinc.UUCP>; the figure is his. ]

Show solution

First a little geometry lesson.  Don't just sit there, sketch this.
You will need a figure.  Let X, Y, and Z be points on a circle whose
center, W [Editor: "N" in the original, which uses "W" for the center
from here on], is contained in the interior of the triangle XYZ.  It is a
theorem of geometry that if we move Z around on what is currently the
arc XZY, the angle XZY will remain constant.  Call the measure of that
angle "a".  Further geometry states that angle XWY has
measure 2a.  Let V be the midpoint of line segment XY.   We have just
divided isosceles triangle XWY into two right triangles.  Then angle VWY
must have measure a.  Call the length of XY "l".  Hence the length
of segment VY is l/2.  The length of WY is the radius of the circle,
call it "r".  Then it follows from the figure that sin(a)=(l/2)/r=l/2r.
Hence, r=l/(2sin(a)).

What this means is that if we want to work backwards to find the locus
of points, Z, so that two points X and Y a distance of l apart will
form the base of a triangle, and angle XZY measures a, one
simply sets one's compass to length l/(2sin(a)) and draws arcs with
centers at X and Y.  The two arcs will intersect at W.  Then draw the
circle with that same radius with center W.  We will get all possible
points Z so that angle XZY has measure a.

With this construction you can use your three points in pairs as X and
Y.  You can get three very different arcs on which O must lie.  That
determines the coordinates of O graphically.  Using the analytic
geometry equivalent of these operations, you can get the algebraic
coordinates solving the simultaneous equations that tells you the
intersection of the three circles.

-- Mark Leeper <...ihnp4!mtgzz!leeper>