The simplest way to find a parametrization of the plane projective curve $XZ-Y^2=0$. I have to explain to some first year math students that the projective algebraic set $\textbf{Z}(XZ-Y^2)\subset\mathbb P^2_k$ is 
$$V=\{(a^2_0:a_0a_1:a^2_1)\subset\mathbb P^2_k \,:\, \textrm{for}\; a_0,a_1\in k\}$$
It is evident that $V\subseteq \textbf{Z}(XZ-Y^2)$, but which is, in your opinion, the simplest way to show the other inclusion? 

Suppose that $(x:y:z) \in \mathbf{Z}(XZ - Y^2)$. Then we know that $xz - y^2 = 0$, i.e.
$$
y^2 = xz
$$
Since $k$ is algebraically closed by hypothesis $\sqrt{x},\sqrt{z} \in k$, thus
$$
(x:y:z) = (x : \sqrt{x} \sqrt{z} : z) \in V
$$
so $\mathbf{Z}(XZ - Y^2) \subseteq V$ and we're done.
