162. The Polygon Puzzle · Closed, straight, and no crossing lines

Try to break it

Drag any corner around the circle. As long as the sides only meet at the corners, it's a valid polygon. Drag a corner past its neighbour and watch the sides cross — that's a self-intersecting figure, no longer a simple polygon.

How you build it

Make a pentagon.

• Place point A, the first vertex of the polygon.
• Place point B, the second vertex of the polygon.
• Place point C, the third vertex of the polygon.
• Place point D, the fourth vertex of the polygon.
• Place point E, the fifth vertex of the polygon.
• Draw a straight segment AB.
• Draw a straight segment BC.
• Draw a straight segment CD.
• Draw a straight segment DE.
• Draw segment EA to close the polygon.

The proof, step by step

Prove that the figure is a polygon — closed, with straight non-crossing sides.

• All sides must be straight line segments. Curved edges disqualify the shape.
• The figure must be closed. An open chain of lines is not a polygon.
• Sides can only intersect at their endpoints (vertices). Crossing sides in the middle breaks the definition.
• Each vertex connects exactly two sides. This ensures a consistent boundary.

Worked example

Which of the following figures is a polygon?

A polygon is defined as a closed plane figure bounded by straight-line segments that intersect only at their endpoints. Curved sides, open shapes, or self-intersecting sides violate this definition.

• A shape with curved sides.
• An open curve with no endpoints connected.
• A closed figure with straight sides that do not cross. — correct
• A figure where sides cross each other in the middle.