Pick’s Theorem 99
Exercises
8.1 Give a direct proof, without using the results proved in this chapter, that Pick’s theorem holds
for triangles of height 1, whose base is parallel to one of the axes.
8.2 Give a direct proof of Pick’s theorem for squares in the integer lattice, whose diagonals (rather
than sides) are oriented with the axes.
8.3 Can we omit the “simple” qualifier in Pick’s theorem? [Hint: For example, does Pick’s the-
orem hold for degenerate triangles (with zero area) or for such figures as the bow tie or the
annulus, shown below?]
8.4 Prove that every simple polygon in the integer lattice can be triangulated by triangles whose
sides contain no lattice points. [Hint: Using lemma 68, it suffices to consider triangles. Argue
by induction on the total number of interior and boundary points. If you have got a triangle
with a boundary vertex on a side, then make two smaller triangles and use the induction
hypothesis.]
8.5 Prove statement (2) of the key lemma 66.
8.6 Does the proof of the key lemma (lemma 66) work in the case that the common part of P and
Q is a single vertex, rather than a side? Prove your answer.
8.7 Is the key lemma true when the common boundary of P and Q is not connected? For example,
perhaps the figures are joined by two or more connected common boundaries simultaneously.
[Hint: Consider exercise 8.3.]
8.8 Can you construct a triangle formed by vertices of the integer lattice with area exactly 3.25?
8.9 Prove that every rectangle formed by vertices in the integer lattice (not necessarily oriented
with the axes) has an area that is an integer.
8.10 In three dimensions, consider the tetrahedron with vertices at (0, 0, 0), (1, 0, 0), (0, 1, 0), and
(1, 1, n) for arbitrary integer n. Draw the tetrahedron, and using the formula for the volume of
a cone over any planar figure,
volume =
1
3
(area of base) ·height,
show that the volume is n/6. Prove that there are no interior lattice points and no lattice points
on the boundary, other than those four. Conclude that there is no three-dimensional analogue
of Pick’s theorem expressing the volume of a polyhedron solely in terms of the number of
interior and boundary vertices.