This is an exact equivalent of my question, can someone re-write the accepted answer in more simplified terms/notation? i.e what does gamma represent? etc..
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Join the line from $A$ to $C$ and let angle $BAC=\beta$. Let the direction of particle $C$ be defined by the angle $\alpha$ it makes with the line $CA$. In this set-up, the angle $\beta$ and the speeds $U_1$ and $U_2$ are known, but the angle $\alpha$ needs to be determined.
In order for the particles to meet, they must be at the same perpendicular distance from the line $AC$ at the same time. Therefore the components of velocity perpendicular to the line $AC$ must be equal and therefore we solve for $\alpha$ the equation $$U_1\sin\beta=U_2\sin\alpha$$.
As was pointed out by @RoryDaulton, there are often two solutions for $\alpha$ depending on the other numbers involved, and obviously there may be situations where interception is impossible - i.e. no real solution for $\alpha$.
David Quinn
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Could you also show how we then get to the coordinates of the intersection point? – RandomUser Feb 10 '18 at 11:09
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@RandomUser when you have the value of $\alpha$ you can use simple trigonometry to find the distance to the intersection point and hence the coordinates – David Quinn Feb 10 '18 at 12:13