2. First, connect the tangent point and center of two circles respectively, and then make the center of the small circle perpendicular to the connected radius in the big circle. You will get a right triangle, the hypotenuse =2+6=8, the long right angle side is equal to the length of the outer tangent, and the short right angle side =6-2=4. According to Pythagorean theorem, the long right-angled side is four times that of root number three, that is, the circumscribed side is four times that of root number three. In that right triangle, the included angle can be found to be 30 degrees according to the value of trigonometric function.
3. According to the tangent length theorem, EP=EB and EP=EA, so EB=EA= 1/2AB, so EP= 1/2AB.
Similarly, FP= 1/2CD, then EF=EP+FP= 1/2(AB+CD). Calculate AB=CD= double base r 1+r2 by the method of the previous question, then EF= double base r 1+r2.
Skip a little, I believe you can understand it at your level! The most important problem of this kind is the auxiliary line, which is often added when solving round problems. I hope you can remember them. I wish you progress in your study! (Remember to give me some points ~)