solves for intersection of angle specified ray with line between two points
uses barycentric coordinate formula, vector algebra
angle specified ray intersection with line between 2 given points
use barycentric coordinate formula of line
scalar b, vectors u, w two points on line to find vector z on line
z = b·u + (1-b)·w
input points A, B create vectors u, w from orgin resp.
create direction vector Θ from angle input α
solve vector equation: a,b scalar, Θ, u, w, z vectors
a·Θ = z = b·u + (1-b)·w
by taking DotProduct with vector perpendicular to Θ
ϕ=PerpendicularVector[Θ]
a·ϕ∙Θ = 0 = b·ϕ∙u + (1-b)·ϕ∙w
slove for barycoordinate b of equation for z
b = ϕ∙w / (ϕ∙w - ϕ∙u)
note that barycoordinate 0<b<1 for z between A, B
equation works up to 180⁰ angle ambiguity
can use the sign of a to resolve the ambiguity
a = Θ∙z

the vector equation, barycoordinate line equation is valid for the whole line
you can add a line through A,B, zoom out and use the angle slider to show z, F tracing the line beyond the interval 0<b<1
GeoGebra's calculation of the areas of the triangles in the ratio view do not preserve the sign, or orientation of the areas
so the area ratio calculation doesn't correctly calculate b for b outside the b [0,1] interval
likewise the line segment length calculation of the barycenter balance shows signs not agreeing outside the b [0,1] interval