<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] For an alternative, more geometric proof, consider that the point of intersection $z$ must lie on the bisector of angle $\angle xoy$ by symmetry, so $z = \lambda(x+y)$ for some $\lambda \in \mathbb{R}\,$.
The power of point $z$ with respect to the unit circle is $|z|^2-1$ and, since $zx$ is a tangent, it also equals $|z-x|^2$.
Therefore, using $\,\bar x = 1/x\,$ and $\,\bar y = 1/y\,$ at the last step, since $|x|=|y|=1\,$:
$$
\require{cancel}
\begin{align}
z \bar z -1 & = (z-x)(\bar z - \bar x) \\
\cancel{z \bar z} -1 & = \cancel{z \bar z} - z \bar x - \bar z x + x \bar x \\
z \bar x + \bar z x & = 2 \\
 \lambda(x+y)\bar x + \lambda (\bar x + \bar y) x & = 2
\end{align}
$$
$$
 \lambda = \frac{2}{x \bar x + \bar x y+\bar x x +x \bar y} = \frac{2}{2 + \frac{x}{y}+\frac{y}{x}} = \frac{2xy}{x^2+y^2+2xy} = \frac{2xy}{(x+y)^2}
$$
Then $z = \lambda (x+y)=\cfrac{2xy}{x+y}\,$.
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