Why are these lines tangent? I was trying the problems at < and for level 20, ending up using, but couldn't see the reason behind the following: !enter image description here We have a circle centred on **B** and a point **A** outside the circle. Construct **E** at the midpoint of **AB**. Draw a circle centred on **E** passing through **A** and **B**. **F** and **G** are the intersection points of the two circles. Now the lines **AF** and **AG** are tangents to the original circle. How can we see that is the case?

Since $AB$ is the diameter of the 2nd circle, $\angle AGB=\frac{\pi}{2}$. This means that $AG \perp BG$ which is the radius of the 1st circle.