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.

Here’s what it looks like.

Hope that helps!

- Uncle Colin