Changing the Border
This is a nifty puzzle whose solution is based on the area of a triangle. The land is divided into two parts (P and Q) by the border ABC. Redraw a straight border line passing through point A so that each of the areas does not change.
http://www.ies.co.jp/math/java/geo/tochia/tochia.html

Dividing a Quadrilateral
This is a more challenging problem that is based on the same underlying concept.
http://www.ies.co.jp/math/java/geo/shi2to/shi2to.html