# Ask Uncle Colin - A Tangential Proof

Dear Uncle Colin,

I have figured out a construction of a tangent line to a circle, but haven’t been able to prove that it works. Can you help? Here’s the protocol:

- Pick two points on the circle, A and B
- Draw a circle centred on B, passing through A
- This circle also intersects the original circle at C.
- Draw a circle centred on A, passing through C.
- This circle also intersects the second circle at D.
- Line CD is a tangent to the original circle.
- Everyone Understands Circles Like I Do

Hi, EUCLID, and thanks for your message!

Here’s the construction, for everyone following along at home.

My reasoning would be:

- Let angle AOC be $2x$
- Then angle BAC = $x$, because AC is perpendicular to OB and OB bisects angle AOC
- Also, angle AOD is $90º - 2x$
- Triangle ACD is isosceles (because AC and AD are radii of the same circle), and has AB as a line of symmetry (CD is a chord of a circle with centre B, which passes through A)
- So angle BAD is $x$ by symmetry
- Angle OAD is $\br{90º-2x} + (x) + (x) = 90º$
- So AD is perpendicular to a radius of the original circle, and is therefore a tangent.

Hope that helps!

- Uncle Colin