Find a point R, such that angle increases 3 times Let $X=(4,0), Y=(4,3), O=(0,0)  $ are points.  I have to find point R with integer coordinates, such that $3|\angle$$XOY|$=$\angle$$|XOR|$. I think it's $R=(-3,8)$, but I am not sure. How can I prove it? Thanks for your help.

If $v, w$ are vectors in $\mathbb{R}^2$, the cosine of the angle between them is equal to $$\frac{v \cdot w}{||v|| \cdot ||w||}$$
Also, the triple angle identity formula can be derived from the formula for $\cos x+ y$: $$\cos 3x = 4 \cos^3 x - 3 \cos x$$
Let $\phi$ be the angle between $X$ and $Y$, and $\theta$ be the angle between $X$ and $R$.  Try using these formulas to see whether $\cos 3 \phi = \cos \theta$.
