Take a piece of paper and put three dots in the middle, about 2 inches apart, to form a triangle with acute angles. (What follows works for any triangle but if you use an obtuse angle the you might find yourself needing a bigger piece of paper.)
Now you have your three points, draw a circle that touches each one of them. How many other circles do you think you could draw which would also touch the three points? It seems intuitively that there ought to be, so have a try if you like but the answer is that there aren’t any. For any three points on a plane (actually three points always make a plane of their own, which is why three-legged stools don’t rock) there is exactly one circle which will touch all three.
The reason is very simple. For a circle to touch just two points (call them A and B), the centre of that circle must fall on an imaginary line that runs perpendicular to the line between A and B, through its middle. Every point on that imaginary line is equal in distance from A and B, so every point on that imaginary line can be the centre of a circle which will touch A and B. As a result, there are an infinite number of circles which will touch A and B.
But what happens when we add a third point, C. We can now make a pair using C and one of the originals – it doesn’t matter which so we’ll take A and C. As above, there’s an imaginary line which runs perpendicular to a line between A and C, again through the middle, with every point on that imaginary line being the centre of a possible circle touching A and C.
So we have two straight lines, which can’t be parallel to each other if A, B and C form a triangle rather than a straight line, so those two lines must cross somewhere. They must cross in one place, and one place only. And that one place is the only point which is equally distant from A, B and C, so it’s the only possible centre for a circle touching all three. We can go further and make a perpendicular line for the remaining points, in our case B and C, which will contain all the points equally distant from B and C, so all the points which can the centre of a circle touching B and C. There will be one point on that line where it crosses the other two lines and it will be precisely where they already meet, since that crossing too has to be the centre of the circle touching the three points.
That’s it. That’s the proof in all its beautiful simplicity. Another example of the way in which mathematics so often cuts through the fog of intuition and apparent complexity to reveal the sometimes simple principles underlying the universe.
In this design, I’ve placed three different triangles in a circle and drawn in the lines perpendicular to their sides. They illustrate the fact that although there’s only one circle touching the three points of a triangle, the converse is that there are an infinite number of triangles that will fit into any circle and all the lines perpendicular to the sides of those triangles meet at the centre of the circle.