The Solution

	Triangles: The Base Case
	Convex Quadrilaterals     Non-Convex Quadrilaterals*     Self-Intersecting Quadrilaterals*
	Convex Pentagons
	Convex Hexagons And Up     The General Case*

Pedagogy

	Math 104     Original Discussion
	Oriented Area Part 1
	Oriented Area Part 2
	Vector Notation

Java Demonstrations

	Area of Midpoint Triangles
	Area of Midpoint Quadrilaterals
	Area Max for Convex Pentagons
	Infinite Area Ratios
	Star Pentagons
	Non-Midpoint Polygons

Bibliography

This page is intended to explain the notation used in the analytic derivation of the area formula for midpoint polygons.

We identify vectors and points, for example (x, y) = X, as illustrated below:

Figure 6. The vector X is identified with the point (x, y).

Now the formula for the area of a triangle can be expressed in vector operations on the vectors associated with the triangle's vertices. Consider a triangle with one vertex at the origin and its other vertices at (x₁, y₁) = X₁ and (x₂, y₂) = X₂ (see figure 7). The oriented area of this triangle is given by the dot product of (1/2)(X₁ x X₂) with (0, 0, 1), as explained in the first page on oriented area. We do not usually write in the dot product, but leave the area expressed as a vector, since this does not change the calculations.

Figure 7. The area of a triangle with one vertex at the origin, expressed in vector notation.

It should be emphasized that the use of vectors here is just a notational convenience, and is not required for the derivation of the formula: it would be possible, if somewhat laborious, to arrive at the same conclusion from calculations involving solely the coordinates of the points.