Ancient Greek mathematicians played by very strict rules when drawing shapes. No marked ruler, no protractor — only an unmarked straightedge (which can only draw straight lines) and a compass (which draws circles and can carry the distance between two points elsewhere). With just these two tools, there are exactly two things you're allowed to do: draw a straight line through two already-marked points, and draw a circle centered on an already-marked point with a radius equal to the distance between two other marked points.
Even with such limited tools, you can create a surprising variety of exact shapes. The classic example is the perpendicular bisector of a segment. Draw two circles of radius AB, one centered at each endpoint of segment AB — these two circles meet at one point above the segment and one point below it. Connect these two intersection points, and remarkably, that line passes exactly through the midpoint of AB while meeting it at a perfect right angle (90°). Without ever measuring a length with a ruler or an angle with a protractor, you've made a perfect perpendicular bisector purely from the property that "two circles of equal radius meet at this spot."
The angle bisector works on a similar principle. Mark two points that are the same distance from the vertex, one on each side of the angle, then draw circles of equal radius centered on each of those points — they meet at a new intersection point. Connect the vertex to that intersection point, and you get a line that splits the original angle exactly in half. Once again, without measuring any angle at all, you've created an exact bisection purely from the symmetry of "equal distance, equal radius."
Constructing an equilateral triangle is even simpler. If you want segment AB to be one side, just draw circles of radius AB centered at A and at B. The point C where the two circles meet is, by definition, a distance of AB from both A and B, so triangle ABC must be an equilateral triangle with all three sides equal to AB.
Repeating this equilateral-triangle idea lets you construct a regular hexagon too. Draw a circle, start from a point on it, and keep drawing circles of the same radius, walking around the circumference — you end up with exactly 6 points marked on the circle. That's no coincidence: the side length of a regular hexagon is always exactly equal to the radius of its circumscribed circle — the very same fact behind six equilateral triangles fitting together to form a regular hexagon.
Straightedge-and-compass construction has its limits too. Somewhat more complicated constructions, like the regular pentagon or octagon, are possible, but it's been mathematically proven that a regular heptagon (a regular polygon with 7 sides) can never be constructed with just these two tools, no matter how hard you try. Whether a regular polygon can be constructed depends on whether its number of sides can be written as a power of 2 times a product of "Fermat primes" (special primes like 3, 5, 17, 257, and 65537), and 7 doesn't satisfy that condition. It was also proven in the 19th century that "trisecting an arbitrary angle exactly" and "drawing a square with the same area as a given circle" are both impossible with just these two tools. It's one of the fun things about mathematics — that even "this can't be done" can be proven rigorously.
On our activity page, first watch the four construction processes unfold step by step in "View Example," then switch to "Construct It Yourself" to pick up the compass and straightedge and construct the perpendicular bisector, angle bisector, and equilateral triangle on your own by clicking. Construct it correctly, and a success message appears right away.