00:02
We are asked to prove picks theorem using strong induction.
00:11
So picks theorem says that the area of a simple polygon p in the plane with vertices that are all lattice points equals ip plus bp over 2 minus 1.
00:25
So we have that ip is a number of lattice points in the interior of a polygon p.
00:41
And we have that bp is the number of lattice points on the boundary of p.
01:04
So to prove the statement first, okay, let's identify what the statement is.
01:11
So pn is the statement, the area of a simple polygon p with vertices that are all lattice points, is going to be ip plus bp over 2 minus 1.
02:17
And of course, this only makes sense if n is going to be.
02:19
Be greater than or equal to three.
02:22
Otherwise, we don't really have something with area.
02:26
So for the base step, now we're giving the hint that for the basis step, we should actually start with theorem for rectangles and then do it for right triangles, and then for all triangles.
03:08
So our basis step is actually n equals three.
03:14
But first, let p be a rectangle with vertices, a, c, b, d, a, d, a, d, b -c, so the way i think of it is sort of like this.
04:01
Got a -c down here, and then you've got a -d over here.
04:15
Up here, you've got b -d, and then you've got b -c.
04:24
Or i guess you could flip those around.
04:26
This is the x -axis, and this is the y -axis.
04:31
Anyways, we have that the width of the rectangle is a -minus b, and the length of the rectangle is going to be d minus c, or i guess you could say c minus d.
05:03
And so it follows that the area of the rectangle is the product of the lithe and length.
05:12
And so area of p is going to be a minus b times c minus d.
05:23
And now since the vertices are acbd, ad, and bc, there are a -c -b -c, there are a -m -m -s -a -a -1.
05:34
B plus 1 lattice points on each side, i guess you'd say each horizontal side, and there will be c minus d plus one lattice points on each vertical side.
06:29
But now we've over counted by two, since in counting the lattice points on the vertical side, we've also counted at both ends, lattice points that lie in the horizontal.
06:43
Sides.
06:44
So this is actually going to be c minus d plus 1 minus 2, or c minus d minus 1 minus points in each vertical side.
06:53
So there are going to be 2 times a minus b plus 1 plus 2 times c minus d minus 1 points in the boundary.
07:21
And we have that, again, since the vertices are ac, b, d, ad, b, c, there are going to be a minus b minus 1 times c minus d minus 1 interior lattice points so to do this i simply took off the lattice points which are on the boundary and so it follows that ip plus bp over 2 minus 1 is equal to a minus b minus 1 times c minus d minus 1 plus the by this by 2, we get a minus b plus c minus d minus 1.
08:50
And this in turn is going to be equal to.
08:53
Well, we have a minus b times c minus d.
08:57
And then we can also subtract an a minus b and subtract a c minus d.
09:01
So this will be a minus b times c minus d plus 1 minus 1, which is just the area of p.
09:13
So we've shown it's true in general for rectangles.
09:17
And now, it wants to show that it's going to be true for right triangles...