How is geometry defined using ZFC?

Basically, to develop "formally" a geomety you have two ways; call them analytic and synthetic respectively.

Analytic

This is our "good old" Analytic geometry :

a point in the space is a ordered triple of real numbers : $(x_1,x_2,x_3)$

a line is the totality of points $(x_1,x_2,x_3)$ such that $u_1x_1 + u_2x_2 + u_3x_3 = 0$, where at least one $u_j (j = 1,2,3)$ is different from zero

and so on ...

But real numbers are definable in set theory; thus - in principle - you can translate into set-theoretic notation the equation of the line.

Synthetic

See Edwin Moise, Elementary Geometry from an Advanced Standpoint (3rd ed - 1990), page 43 :

space will be regarded as a set $S$; the points of space will be the elements of this set. We will also have given a collection of subsets of $S$, called lines, and another collection of subsets of $S$, called planes.

Thus the structure that we start with is a triplet : $<\mathcal S, \mathcal L, \Pi>$, where the elements of $\mathcal S, \mathcal L, \Pi$, and are called points, lines and planes, respectively.

Our postulates are going to be stated in terms of the sets $\mathcal S, \mathcal L$, and $\Pi$.

Here are the first two postulates :

I-0 : All lines and planes are sets of points.

I-1 : Given any two different points, there is exactly one line containing them [we can "trivially" express the fact that the point $Q$ is contained into the line $l$ with the formula : $Q \in \mathcal S \land l \in \mathcal L \rightarrow Q \in l$ ].

We write $\overline{PQ}$ for the unique line containing $P$ and $Q$.

We define the relation of betweenness between (sic !) three points $P, Q, R$.

Then [see pages 64-65] : if $R,Q$ are two points, the segment between $R$ and $Q$ is the set whose points are $R$ and $Q$, together with all points between $R$ and $Q$.

The ray $\overrightarrow {AB}$ is the set of all points $C$ of the line $\overline {AB}$ such that $A$ is not between $C$ and $B$. The point $A$ is called the end point of the ray $AB$.

An angle is the union of two rays which have the same end point, but do not lie on the same line. If the angle is the union of $\overrightarrow {AB}$ and $\overrightarrow {AC}$, then these rays are called the sides of the angle; the [common] end point $A$ is called the vertex.

Finally, you can "close the circle" between this two approaches.

Assuming that we have defined the set $\mathbb N$ of natural numbers inside set theory [but I prefer to say that we have defined a model of the natural number system], and then the set $\mathbb R$ of real numbers, we can use $\mathbb R^3$ and call it : (three-dimensional) space.

Comment

What have we gained so far ? I think nothing more and nothing less than what we already have with Descartes' discovery of analytic geometry : an "embedding" of the euclidean geometry into the "cartesian plane".

Of course, the "basic" set-theoretic language gives us a powerful tool for expressing also geometrical "facts" : we can write $P \in l$ for : "the point $P$ is contained into line $l$", we can write $l_1 \cap l_2 \ne \emptyset$ for "two lines intersect each other", ...

But I think that speaking of "foundation for most of modern mathematics" can be mesleading.