Distances to Vertices
In the square below, points that are closer to the center than one of the vertices is coloured yellow. The area of the yellow region is 1/2 of the area of the square.
Area-Bisecting Lines
Every line going through the center of a square bisects its area. Does every line going through the center of an equilateral triangle bisect its area too?
Circle-Bisecting Lines
All straight lines that bisect the area of a circle go through its center.
Folding Squares
If we draw a line through the center of a square, we can fold the square in half along this line. Depending on the angle this line makes with the horizontal, as shown in the diagram, the folded square will have different looks.
Point in a Polygon
For any point inside a square, the sum of the distances from this point to the four sides is half of the perimeter.
Squares in Squares
Let’s draw a smaller square inside a unit square by dividing out a fixed fraction of each side and connecting the dividing point to an opposite angle.