Calculate point coordinates from other points

Question

As in the image below i have four points.

$P_1,P_2,P_3$ are known distinct points ( i know the $x,y$ of each of them )

also the angles $a_1,a_2$ are known.

Can i calculate the coordinates of $x,y$ of $M$ with only this information ?

Answer 1

Suppose $P_1,P_2,P_3$ are not collinear, so lie on a unique circle $C$ with center $O.$ Then let $a_1$ be the central angle in circle $C$ determined by points $P_1,P_2$ (so the arc of the circle with those endpoints which does not include $P_3.$) Similarly let $b_1$ be the central angle determined by points $P_2,P_3$ (the circle arc with those endpoints not including $P_1.$)

If your $M$ was uniquely determined given any three noncolliner points and any two angles $a,b$ then $M$ would be determined by the specific angles $a=a_1/2,\ b=b_1/2.$ However any point $M$ on the arc from $P_3$ to $P_1$ (having those ends and not including $P_2$) will make those same angles $a,b$ which you define, since an angle inscribed in a circle is half the central angle subtended.

This is in a way a quite special case which must be excluded, but it shows nonuniqueness of $M$ in case the $P_k$ are noncollinear. I haven't thought about the collinear case.

Edit: In the collinear case, where each $P_k$ lies on one line $L,$ clearly the point $M$ must be not on $L$ in order for the angles $a,b$ to be defined. When any such $M$ is chosen, and angles $a,b$ found, the point $M'$ obtained on reflecting $M$ through $L$ will make those same angles $a,b.$ So even in the collinear case the data do not determine $M$ uniquely. On the other hand, if one restricts $M$ to one of the two open halfplanes determined by $L,$ it seems clear $M$ is uniquely determined, though a geometric/algebraic way to find it escapes me [use of cosine law e.g. leads to a mess].

Answer 2

The red arcs are the locus of points from which $P1$ and $P2$ are separated by an angle of $a1$. The green arcs are the locus of points from which $P2$ and $P3$ are separated by an angle of $a2$.

Therefore, at each of the points $M$, $N$, $Q$, and $R$, the points $P1$ and $P2$ are separated by an angle of $a1$ and the points $P2$ and $P3$ are separated by an angle of $a2$. The order of the points change, but the angles between them are as given.