A ruler without marks and a compass: with just these two tools, the Greeks constructed midpoints, perpendiculars and parallels with total precision.

Who discovered

The classical constructions are in The Elements of Euclid (c. 300 BC): the perpendicular bisector and the midpoint (proposition I.10), the angle bisector (I.9), the perpendicular by one point (I.11 and I.12) and the parallel by one point (I.31). The rule of using only a ruler without marks and a compass comes from Euclid’s first three postulates: draw a straight line through two points, extend it and draw a circle with any center and radius.

The story

For the Greeks, a construction was a demonstration in the form of a drawing: each step needed to be justified by already proven results. That’s why a graduated ruler wasn’t worth it — measuring is approximating, building is guaranteeing.

Three famous problems have resisted these tools for more than two thousand years: dividing any angle into three equal parts, constructing a cube with twice the volume of another and constructing a square with the same area as a circle. In the 19th century, Pierre Wantzel (1837) and Ferdinand von Lindemann (1882) proved that the three are impossible with just a ruler and compass.

In everyday life

  • Carpentry and metalwork. The dry point compass marks centers, divides pieces in half and draws perpendiculars without needing a square.
  • Technical drawing. The “perpendicular bisector”, “angle bisector” and “perpendicular” tools of CAD programs and GeoGebra repeat Euclid’s constructions.
  • Gardening. With two stakes and a piece of string, you can mark a circular flower bed or a line perpendicular to a wall.

The four constructions

In each figure, drag the points and advance one step at a time. Repeat the construction on paper, with a real ruler and compass.

perpendicular bisector and midpoint

Definiçãoperpendicular bisector

The perpendicular bisector of a segment $\overline{AB}$ is the line perpendicular to $\overline{AB}$ that passes through its midpoint. Every point on perpendicular bisector is at the same distance from $A$ and $B$.

Construction of perpendicular bisector: points P and Q are at the same distance from A and B.

Why does it work? The arcs have the same radius, so $PA = PB$ and $QA = QB$. Points at the same distance from $A$ and $B$ form perpendicular bisector; as $P$ and $Q$ are in it, the straight line $PQ$ is perpendicular bisector.

angle bisector of an angle

Construction of angle bisector: triangles OPR and OQR are congruent (case LLL).

Why does it work? $OP = OQ$ (first arc) and $PR = QR$ (second pair of arcs); the $OR$ side is common. In the LLL case, the triangles $OPR$ and $OQR$ are congruent, and the angles in $O$ are equal.

Perpendicular to a point outside the line

Perpendicular through P: points P and E are both at the same distance from C and D.

Why does it work? $PC = PD$ and $EC = ED$: $P$ and $E$ are on perpendicular bisector of $\overline{CD}$, which is perpendicular to the line $r$.

Parallel by a point

Parallel by P: the four sides of CDEP are equal, so CDEP is a rhombus.

Why does it work? The four sides of $CDEP$ have the same measurement: it is a rhombus, and every rhombus is a parallelogram. Therefore $PE$ is parallel to $CD$, that is, to the straight line $r$.

Check the answer

45°: construct a perpendicular (90°) and then the angle bisector of this right angle. 60°: with the dry tip at A and opening AB, draw an arc; repeat with the dry end at B. The meeting point C forms equilateral triangle ABC, whose angles measure 60° — exactly the first proposition of the Elements.

True or false?

  1. Every point on perpendicular bisector of AB is at the same distance from A and B.
  2. With a ruler and compass, you can divide any angle into three equal parts.
  3. The angle bisector of a right angle forms two 45° angles.
  4. In Greek constructions, the ruler could have centimeter marks.

Sources: Elements, book I (D. Joyce) · Wantzel (MacTutor) · Lindemann (MacTutor).