Hello, lego_lax!

If no one is responding, it's that we didn't understand the description.

After a number of incorrect sketches, I finally caught on.

I have $\displaystyle \Delta ABC$ inscribed in a circle with $\displaystyle \angle A \:=\:60^o$.

The height relative to BC is equal to the radius.

On the line BA take the segment AD equal to AC.

Prove that the line DC is tangent to the circle. Code:

D
*
/
/
/
/
A/
* o *
* / \* *
* / \ * *
* / \ * *
/ r\ * *
* / \ o C
* / o *
* / * E *
/ *
B o *
* *
* *
* * *

We have inscribed triangle $\displaystyle ABC$, with $\displaystyle \angle BAC = 60^o.$

$\displaystyle AE \perp BC,\;\;AE = r.$

Extend $\displaystyle BA$ to $\displaystyle D$ so that $\displaystyle AD = AC.$

Prove that $\displaystyle DC$ is tangent to the circle.

Now I can give it a try . . .