<s>[INST] Given $|x|=|y| = 1$, show that the intersection of two tangents to a unit circle at points $x$ and $y$ is $\frac{2xy}{x+y}$. 
Given $|x|= |y| = 1$, show that the intersection of two tangents to a unit circle at points $x$ and $y$ is $\frac{2xy}{x+y}$.

What would be the best place to start to attempt this problem?
What I know:
$x^2= x\overline{x} = 1 = y\overline{y}$
 [/INST] Let $Z $ be their intersection point. Then from here, we know that $$z +x^2\bar z =2x \text {   and  } z+y^2\bar z =2y $$So $$(x^2-y^2)\bar z=2 (x-y)$$ $$\Rightarrow \bar z =\frac {2}{x+y} $$ Thus $z = \frac {2}{\bar x + \bar y} $. Since $x $ and $y $, lie on the unit circle, $$\bar x =\frac{1}{x} \text {    and   } \bar y = \frac {1}{y} $$ and the result follows. Hope it helps. 
</s>