Determine if a Point is inside a polygon without coordinates I have a polygon $ABCD$. I do not know the coordinates of the corners but I know the length of its sides (i.e. I know length of $AB$, $BC$, $CD$ & $DA$).
I have a point $P$. I do not know the coordinates of this point either but I know the distances $AP$, $BP$, $CP$ & $DP$.
How do I determine if point $P$ lies inside the polygon $ABCD$?
 A: With $P$ as center draw four circles $\gamma_i$ $(1\leq i\leq4)$ with radii $|PA|$, $\ldots$, $|PD|$. Choose $A_1$ on $\gamma_1$ arbitrarily and construct a point $A_2 \in \gamma_2$ such that $|A_1A_2|=|AB|$. There are two such points, by symmetry you can discard one of them. Then construct the two points $A_{3i}\in\gamma_3$ having  distance $|BC|$ from $A_2$, and keep both of them. Finally construct the four points $A_{4ik}\in\gamma_4$ having  distance $|CD|$ from $A_{3i}$. At least one of these four points should have  distance $|DA|$ from $A_1$. This allows you to draw at least one quadrangle $Q$ satisfying the given conditions. Now check whether $P\in Q$.
Of course you can do all of this numerically as well: Put $P=(0,0)$, $A_1=\bigl(|PA|,0\bigr)$, and proceed in terms of analytic geometry: intersection of circles, etc.
A: Try this: You have the Distance AP. Now you take AB & DA to get to the same distance ($x*AB +y*DA = AP$). Repeat that for every corner point. x and y have to be both smaller than 1 to have the point in the polygon.
