First, you draw two common tangents, make an intersection point P, and make a connecting line between the two centers. Then you can prove that this line must pass through point P. Then you can make four vertical lines from the two centers to the two common tangents, and you can see four right-angled triangles. Then you can prove that the triangles are similar. Then, by using the similarity, it can be calculated that Po 1 is 4 and Po2 is 6.

Second, make the center line, an inner common tangent, and then tangent the two centers to the inner common tangent to get two right triangles, which can prove that they are similar. Because the inner common tangent divides the center line into two segments, the lengths of the two segments can be obtained by proportional similarity (3.75 and 6.25, namely 15/4 and 25/4, respectively). Then, using Pythagorean theorem, the answer is more troublesome. I didn't count.

The third question is relatively simple. Because there is only one common tangent between P and circle O, it must be inscribed. Connecting the two centers, we can calculate that the radius of circle P is 8-5=3.

Finally, the number one problem is that the connecting line of the outer circle must pass through the intersection of two internal common tangents. But not necessarily equally divided, if and only if the radii of two circles are equal.