How to check if a point is inside a rectangle? There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?


Given how much attention this post has gotten and how long ago it was asked, I'm surprised that no one here mentioned the following method. 
A rectangle is the image of the unit square under an affine map. Simply apply the inverse of this affine map to the point in question, and then check if the result is in the unit square or not.
To make things clear, consider the following image,

where the vectors in the pictures are $\mathbf{u} = c - d$, $\mathbf{v} = a - d$, and $\mathbf{w} = d$.
Since the legs of a rectangle are perpendicular, the matrix $\begin{bmatrix}\mathbf{u} & \mathbf{v}\end{bmatrix}$ is orthogonal and so we even have a simple formula for the inverse:
$$\begin{bmatrix}\mathbf{u} & \mathbf{v}\end{bmatrix}^{-1} = \begin{bmatrix}\mathbf{u}^T/||u||^2 \\ \mathbf{v}^T/||v||^2\end{bmatrix}.$$
If you want to check many points for the same rectangle, this matrix can be easily precomputed and stored, so that you perform the (typically more expensive) division operations only once at the start. Then you only need to do a few multiplications, additions, and subtractions for each point you are testing.
This method also applies more generally to checking whether a point is in a parallelogram, though in the parallelogram case the matrix inverse does not take such a simple form.
