# Drawing a tangent to a circle at a given point in just 3 ruler-and-compass constructions

A friend of mine recently introduced me to an interesting app called Euclidea that challenges you to complete geometric constructions like the ones we learnt in high school. The app has successfully made me feel quite silly about myself, for I am unable to solve this elementary problem:

Given a circle and a point on its circumference, construct the tangent to the circle at that point using ruler-and-compass constructions.

The difficulty I am facing is in finding an optimal solution: I want to accomplish this using as few constructions as possible. So, drawing a line with the ruler would count as one construction, and drawing a circle with the pair of compasses would count as one construction.

I am able to do it in four constructions like this:

But, apparently there is a solution using just three constructions. And, after several weeks of trying (and failing), I have decided to ask for help. Can anyone help me see the light?

1. Pick an arbitrary point $A$ on the circle, close to $P$, the point of tangency. (0 moves)
2. Draw a circle centred at $A$ passing through $P$ and intersecting the circle again at $B$. (1 move)
3. Draw a circle centred at $P$ passing through $B$. Let it intersect the circle centred at $A$ at $C$. (2 moves)
4. Draw $PC$. (3 moves)