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Geometric constructions
Just a straight edge and a compass. No measuring. The classical Greek tradition — and a surprising amount of what's possible.
Bisect a segment — ruler & compass
The classical construction. Click through.
step 1 / 4
Step 1
Start with a line segment AB.
Why it works
Both crossings are equidistant from A and B — so they sit on the perpendicular bisector.
Two tools, no rulers
Classical construction allows just two operations:
- Draw a straight line through two given points (straightedge).
- Draw a circle with a given center, passing through a given point (compass).
That's it. No measuring lengths or angles. Despite the restriction, you can do an enormous amount.
What you can construct
- Bisect any segment or angle.
- Drop or raise a perpendicular.
- Copy any segment or angle.
- Construct equilateral triangles, squares, regular hexagons, regular pentagons.
- Inscribe a circle in any triangle (incircle).
What you can't
Three famous problems were proved impossible with ruler and compass alone:
• Trisect a general angle.
• Double the cube (find side of cube with twice the volume).
• Square the circle (find a square with same area as a given circle — needs π).
• Trisect a general angle.
• Double the cube (find side of cube with twice the volume).
• Square the circle (find a square with same area as a given circle — needs π).
Why bother?
Constructions are the original proofs. Showing you can build something with these two tools is showing it logically follows from the axioms — no measurement, no approximation. The whole tradition of geometric proof grew out of this discipline.