Level 17: Given a point A, a line, and a point B. Construct a circle that passes through A and is tangent to the line at B.

Hint: The circle passes through A and B, what else?

Solution: The circle must also pass through the mirror image of A. The line is a tangent, which means the line through B perpendicular to it is the diameter. The point D, the mirror image of A, can be constructed by first constructing a parallel line through A parallel to the tangent, and intersect it with the circle with center B and radius AB. The rest? It is just level-15, the circumcircle of A, B and D!

For economy we could have just constructed the perpendicular bisector of AB and then the line through B perpendicular to the tangent and use that as the circumcenter, but it would be hard to explain why that’s true.