A Voronoi partition is made by taking a set of sites spread on the screen and drawing boundaries between the sites, in such a way that each boundary is exactly halfway between the sites on either side of it.

Let's start with two sites, $A$A and $B$B. Every point on the line that divides them, is an equal distance from $A$A and from $B$B. What other properties does this line have?

The first thing we can do is to draw a line segment from $A$A to $B$B. Since all the points on the boundary are an equal distance from $A$A and from $B$B, the boundary will cross $\overline{AB}$start overline, A, B, end overline at the midpoint, $M$M.

You can learn more about midpoints here.

Now we can pick another point on the boundary, $P$P and make two triangles: $\triangle AMP$triangle, A, M, P and $\triangle BMP$triangle, B, M, P.

Point $P$P is on the boundary so we know that $\overline{AP} = \overline{BP}$start overline, A, P, end overline, equals, start overline, B, P, end overline.
We also know that $\overline{AM} = \overline{BM}$start overline, A, M, end overline, equals, start overline, B, M, end overline and that both triangles share side $\overline{MP}$start overline, M, P, end overline.

Since both triangles share three side lengths, they are similar triangles. You can learn more about similar triangles here.

Since $\triangle AMP$triangle, A, M, P and $\triangle BMP$triangle, B, M, P are similar we know that $\angle AMP = \angle BMP$angle, A, M, P, equals, angle, B, M, P. Since both angles lie on a straight line (line $AB$A, B), we know that $\angle AMP + \angle BMP = 180^\circ$angle, A, M, P, plus, angle, B, M, P, equals, 180, degrees. So both angles must be $90^\circ$90, degrees.

Therefore, the boundary between $A$A and $B$B is the perpendicular bisector of the line $AB$A, B. You can learn more about perpendicular bisectors here.

So we know the boundary between two sites is a perpendicular bisector of the line joining those sites. How can we use this information to find the Voronoi partition of three sites? First let's start by drawing lines between the sites.

Next we find the midpoints of the edges and draw the perpendicular bisector for each edge.

Notice that the three perpendicular bisectors meet at a point, $V$V. Since each perpendicular bisector is an equal distance from two sites and $V$V lies on all three of them, $V$V must be an equal distance from all three site. $V$V is therefore a vertex in our Voronoi partition. We can draw our final boundaries by starting from this vertex and continuing through each of the midpoints.

Now we've seen some of the geometry of a simple Voronoi partition, let's use algebra to calculate the coordinates for a vertex in a Voronoi partition.

Say we have three sites, $A$A, $B$B, and $C$C with coordinates:

Let's first calculate the equation for the perpendicular bisector of $\overline{AB}$start overline, A, B, end overline. We know this line will pass through the midpoint of $\overline{AB}$start overline, A, B, end overline and have the negative inverse gradient of $\overline{AB}$start overline, A, B, end overline.

Learn how to calculate the equation of a perpendicular line here.

Now we have the equation for the perpendicular bisector of $\overline{AB}$start overline, A, B, end overline, we can use the same approach to calculate the perpendicular bisector of $\overline{BC}$start overline, B, C, end overline.

Now we have the equations for two perpendicular bisectors we can calculate where they intersect. If we use $m_1$m, start subscript, 1, end subscript and $b_1$b, start subscript, 1, end subscript to represent the gradient and $y$y-intersect the first perpendicular bisector and $m_2$m, start subscript, 2, end subscript and $b_2$b, start subscript, 2, end subscript to represent the gradient and $y$y-intersect the second perpendicular bisector, we have:

So:
$\qquad \begin{aligned} m_1 x + b_1 &= m_2 x + b_2 \\ \\ m_1 x - m_2 x &= b_2 - b_1 \\ \\ (m_1 - m_2)x &= b_2 - b_1 \\ \\ x &=\dfrac{b_2 - b_1}{m_1 - m_2} \end{aligned}$

Plug the values for the gradients and $y$y-intersects into the equation to find the $x$x coordinate of the intersection point. Then plug the value for $x$x into one of the equations for a perpendicular bisector to find the $y$y coordinate.

Now imagine calculating this for hundreds of sites. This is why we use computers!