How to find the distance between a point (x1,y1) and a line segment between (x2,y2) and (x3,y3) measured at an angle ( say theta ) ? How to determine if the line segment can be reached from the point at the given angle?
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Answerer 1
Label the points A(x₁,y₁), B(x₂ ,y₂), C(x₃,y₃). Then angles ABC=u, ACB=v are known quantities (I assume you know how to find tanu and tan v, which is fairly easy). If tanu<tanθ<tanv, the chosen point, X say, can be reached, if not, not. AX can be found from the relations AB²=AX²+BX²-2AB.BXcosu and
AC²=AX²+(BC-AX)²-2AX.(BC-AX)cosv. These are two equations in the unknowns AX and BX, all the other quantities being knowns, so AX can be found from the algebra.