I have been mulling over a really interesting question in analytic geometry that is much harder than it first appears to be. Hope you can provide some insight into solving it.
If you only know:
- The lengths of line segments A, B and C
- The coordinates of the end position of line segment C (X,Y)
Define the internal angles d, e and f in terms of A, B, C and (X,Y).
What I've done to approach the problem:
- Make a right angled triangle from origin, (X,Y) and (X,0) {last point is simply the where (X,Y) meets the x axis}
- Draw an imaginary line from where line segments A and B meet to (X,Y)
- Define angle between A and hypotenuse of the right triangle from (1) using the imaginary line in (2) through cosine rule
- Define angle f using the imaginary line in (2) through cosine rule
- Equate the two equations in (3) and (4) so that the two unknown angles are defined in terms of each other
I am stuck at this point. Am I following the correct approach, or does something else need to be done to define angles d, e and f using only information provided? Thanks!
Is there in fact a unique solution?
Consider a circle, radius $A$, centered at the origin, and another circle, radius $C$, centered at $(X, Y)$ Any line segment, length $B$ with end points anywhere on the two circles, would represent a solution.