A triangle T has vertices O, (3, 0), (2, 3). Then the centre of gravity of T lies at (3/2, 1).
True ; False
False. It should be (5/3, 1). You can obtain this as 1/3[(0, 0) + (3, 0) + (2, 3)].
2.
A triangle S has vertices (–1, –1), (0, 2), (1, –1). Then the centre of gravity of S lies at the origin.
True ; False
True. 1/3[(–1, –1) + (0, 2) + (1, –1)] = (0, 0). Or you might argue that the centre of gravity lies on the axis of the triangle and a third the way up.
3.
Give a form of Winternitz’s Theorem for parallelograms.
As the centre of gravity here is also the centre of symmetry, the ratio of the two parts is always 1.
4.
In Q3, what other sets would give the same result?
Any set which is symmetric about its centre of gravity: circular disk, ellipse, regular hexagon etc.
5.
What would a critical set for Ehrhart’s Theorem look like in the triangular lattice?
Much the same: shear the right-angled Ehrhart triangles left or tight through 30°.
6.
Describe two quite different sets in the plane which have the same width in every direction.
The circle (circular disc is an obvious choice. But there are an infinite number of possibilities. Look up "Reuleaux triangle" on the internet.
