Coordinate geometry
If two points are known then the midpoint between them is easily found.
Suppose one point A has coordinates (a,b) and the second point B has coordinates (c,d), then the midpoint M has coordinates:
x=(a+c)/2 and y=(b+d)/2
Each coordinate is the mean average of the coordinates of A and B.

Clearly, if two points can be plotted on a graph then their midpoint can always be plotted too.

Two straight lines either cross - and have one point of intersection - or they are parallel and have no point of intersection.

The situation with a curve and a straight line is more complicated - in fact it is complex.
This diagram (produced by Cinderella) shows a straight line crossing a circle. The points of intersection are shown together with the midpoint of these points. Drag the green line away. Eventually, the two intersection points will disappear but the midpoint remains!

You can move the centre of the circle towards the midpoint and actually make them coincide. If you move the point on the circumference of the circle anywhere you will find that the midpoint will not move at all.

Here's another way of looking at it. This time the intersection of the green line and the line perpendicular to the green line and passing through the centre of the circle is displayed.
Clearly, this intersection will always be visable however the circle and green line are moved.

Move the centre of the circle towards the intersection, ie., along the perpendicular line. You should be able to make this centre and the intersection point coincide. If you move the point on the circumference of the circle anywhere, the intersection will not move at all.

The point is: this intersection and the previous midpoint are the same point!

Where the circle crosses the line - if it does
In order to use algebra to find where the circle and line cross, we need equations for them both and then we have to solve these equations simultaneously. All straight lines have equations of the form y=mx + c.
A circle of radius r and centre (a,b) has the equation (x-a)² + (y-b)²=r²
To find the points of intersection of the line with the curve you have to solve simultaneously the two equations.

So, as you see, this process always leads to a quadratic equation whose roots are the coordinates of the points of intersection. It looks a very complicated equation but the important thing is we know a lot about quadratic equations. We know they always have roots, though sometimes those roots are complex numbers.

If you have installed the plugin (see Tech), find out more about finding the points of intersection.

From the theory of complex numbers, we know that the midpoint of these two intersection points will always exist (and can therefore be plotted) even when the quadratic equation has complex roots.