This example is taken from Silverman and Tate's Rational Points on Elliptic Curves.
Suppose we have the line
$C_1: x+y=2$
and the circle
$C_2: x^2 + y^2 = 2$.
We see that these intersect at the point $(1,1)$. We then put the curves in homogeneous coordinates and see that they are described by
$C_1: X+Y=2Z$ and $C_2: X^2 + Y^2 = 2Z^2$ and we get the single intersection point $(1:1:1) \in \mathbb{P}^2$.
Now, it is said in the text that since the line $C_1$ is tangent to the circle $C_2$ at $(1,1)$, this intersection point "should count double". Can anyone explain the intuition behind this statement?
Intuitively, if we have two irreducible plane curves $F$ and $G$ that intersect at a point $P,$ then the intersection multiplicity $I(P, F \cap G)$ is the maximum number of intersections of $F$ and $G$ near $P$ if we wiggle $F$ and $G$ a little.
For example, take $F$ to be a curve and $G$ to be a straight line that goes through $F$ at $P$ transversally. Then there is at most single point of intersection even if we wiggle $G$ a bit, so the multiplicity of this intersection is $1.$
Now take the cubic with a cusp $F= Y^2-X^3$ and the line $G= X.$ Then $G$ intersects $F$ at $P=(0,0)$ and if we shift $G$ a little to the right, it intersects $F$ in two places so $I(P,F\cap G)=2.$