How do you draw the tangents from a point C to a circle with center A? Find the midpoint D of AC and draw the circle centered at D through C. It crosses the first circle at E and F — the points of tangency. It works because an angle inscribed in a semicircle is a right angle.
The source code is at github.com/KirillOsenkov/LiveGeometry. It is written in .NET and Avalonia.