greenber@timeinc.UUCP (Ross M. Greenberg) (06/23/85)
A puzzle that hit me on the head a few days ago. Given three points (X)
whose co-ordinates are known, and some point within the triangle
formed by these other points, how do you find the location of
the "interior" point given only that the angles through 'O' are known
X (point 1)
|
|
|
O <------ This angle (and the others like it)
/ \ are known
/ \
/ \
/ \
(Point 2)X \
\
X (Point 3)
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Ross M. Greenberg @ Time Inc, New York
--------->{ihnp4 | vax135}!timeinc!greenber<---------
I highly doubt that Time Inc. they would make me their spokesperson.leeper@mtgzz.UUCP (m.r.leeper) (07/01/85)
>A puzzle that hit me on the head a few days ago. Given >three points (X) whose co-ordinates are known, and some >point within the triangle formed by these other points, how >do you find the location of the "interior" point given only >that the angles through 'O' are known > > X (point 1) > | > | > | > O <------ This angle (and the others like it) > / \ are known > / \ > / \ > / \ > (Point 2)X \ > \ > X (Point 3) > 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, N, 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 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 isoceles triangle XWY into two right triangles. Then angle VWY must 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 measures a, one simply sets ones compass to length l/(2sin(a)) and draw 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 of 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