The color lines are parallel to the bases of the trapezoid. The red line passes through the intersection of the diagonals. The green line divides the trapezoid into two similar trapezoids. The yellow line is a bimedian. The blue line divides the trapezoids into two trapezoids with the same area. Prove $$H=\left(\dfrac{\frac1a+\frac1b}{2}\right)^{-1},\quad G=\sqrt{ab},\quad A=\dfrac{a+b}{2},\quad R=\sqrt{\dfrac{a^2+b^2}{2}}$$ and $$\min\{a,b\} \le H \le G \le A \le R \le \max\{a,b\}$$