# The Game of Poncelet

Play the game of Poncelet! Place two circles, red and blue,
set a starting point on the blue circle, and watch what
happens. For more explanations see below.

## What happens?

Starting with a point on the blue
circle, and a tangent to the red
circle containing this point, there are two
elementary operations we may perform.

On the one hand, if a line meets the blue circle, it does so with with
multiplicity 2, meaning that either there are two
intersection points, or if there is only one intersection
point, the line is tangent to the circle. So if already one
intersection point is given, we can replace it with the
other (or do nothing if the line is tangent).

On the other hand, if for a given point there is a
tangent to the red circle, then
either the point lies already on the red
circle, or there is also a second tangent. The second
operation then switches the tangents (or does nothing if the
point is on the red circle).

If we compose these two operations, we obtain the
*Game of Poncelet*, which you can play above. The
question is, can you find a configuration where the game
returns to the starting point?

## How to use the web app

- You can place the circles by clicking three times on
the canvas. Unless your three points lie on a line, there
exists a unique circle containing these three points.
- Once you have placed the blue
circle, click anywhere on the canvas to choose a
starting point, which will automatically be projected on
the blue circle, if you avoid
the centre.
- You do not need to choose the tangent to the red circle, it will be computed
automatically. However, this is only possible if the
starting point lies outside the red
circle, so keep this in mind when setting up the
configuration of the circles.
- Now you can play, and also apply the operation 10
times at once.
- If you wish to try a different configuration, click
the reset button, or start selecting a circle again.