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.
Chord-bisecting Chords
A circle of radius 5 has chord AB with length 6. A second chord is drawn here and is bisected by AB.
Sierpinski Pyramid
Let’s remove the central portion of a regular tetrahedron such that what remains are 4 tetrahedrons each with half of the side length of the original.
Nonlinear Arrangements
Here’s an arrangement of the numbers 1 to 4, and what it looks like plotted into a 4x4 grid such that each row corresponds to a number in the arrangement. Notice that the 3 dots on the bottom right form a straight line.
Nonlinear Dots
Here’s a way to put 6 dots into a 3x3 grid such that no 3 dots form a straight line.
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.
Triangles in a Square
This problem is inspired by this geometry puzzle on Twitter. The original problem asks to find the ratio between two equilateral triangles fitted into a square. Let’s generalize it.