E35 - Three Triangles in a Regular Hexagon

Here is another problem from the Catriona Shearer collection. ABCDEF is a regular hexagon. Lines have been drawn to create three triangles, whose interiors have been shaded. What fraction of the hexagon is shaded?

The points labeled M_i are the midpoints of  their sides, and the unlabeled point marks the center of the hexagon.

E34. Tangent circles in a rectangle

This problem was called to my attention to Apratim Roy. It comes from a collection, on Twitter, of many dozens of interesting geometry problems, at the level of high school geometry, by Catriona Shearer, a math teacher in the UK, which she has posted on Twitter, @Cshearer41.

In the diagram below. you are given circles O and P, externally tangent at B. Circle O is tangent to two sides of rectangle DEFG, and circle P is tangent to three sides of the rectangle, as shown. Find the measure of angle ABC.

It is quite surprising that you don’t need to be given any measurements to answer the problem. (But no fair using that fact in your proof; that is, you can’t just prove a special case such as two circles of equal size.) I also like this problem because it is hard, but not too hard, and because there are several different nice ways to solve it.