Try this:
return new Rectangle((this.ClientRectangle.Width / 2) - 15, this.ClientRectangle.Bottom - 30, 30, 30)
They way you were doing it, the top of the new rectangle starts at the exact bottom of the first one, so when it draw, it goes below the bottom of the first, making it so you can't see it.
The -15 part is to center it in the center of the rectangle. Basically newWidth / 2.