Hi I'm trying to figure out how to calculate the coordinates of a dot at a certain percentage point on an arc. Let's say the dot starts at (800, 300), the half-way points is (400, 0) and the end point is (0, 300). Let's say I want to calculate where the dot will be after traveling 25% of the arc. I'm really clueless where to start on this, thanks for your help.
It appears from the comments that you are happy to take the arc to be an arc of a circle. There is conflicting evidence as to whether you can work out what the circle is, so I'll discuss that. If the radius of the circle is $r$, then, by symmetry, the center of the circle is at $C=(400,r)$. We must have $$(400)^2+(r-300)^2=r^2,$$ an equation which is easy to solve to get $r$.
Now you want the point halfway from $A=(800,300)$ to $B=(400,0)$. You have to find the angle $\theta=\angle ACB$, then you have to bisect it, then you have to find where the bisector meets the arc, that is, you want the point $D$ on the arc for which $\angle ACD=\theta/2$. Come back if you have any trouble carrying out these calculations.