Definition:Construction/Geometric Construction

Definition

Geometric construction is the process of creating a particular geometric figure, according to an appropriate system of axioms.

The unspoken understanding is that those axioms are at base Euclid's postulates and common notions.

Examples

Perpendicular through Point

From Euclid's The Elements:

Proposition $11$ of Book $\text{I} $: Construction of Perpendicular Line

Let $AB$ be the given straight line segment, and let $C$ be the given point on it.

Let a point $D$ be taken on $AB$.

We cut off from $CB$ a length $CE$ equal to $DC$.

We construct an equilateral triangle $\triangle DEF$ on $DE$.

We draw the line segment $FC$.

Then $FC$ is the required perpendicular to $AB$.

Bisection of Angle

From Euclid's The Elements:

Proposition $10$ of Book $\text{I} $: Bisection of Angle

Let $\angle BAC$ be the given angle to be bisected.

Let $D$ be an arbitrary point on $AB$.

From Proposition $3$: Construction of Equal Straight Lines from Unequal, let $AE$ be cut off from $AC$ such that $AE = AD$.

From Euclid's First Postulate, let the line segment $DE$ be constructed.

From Proposition $1$: Construction of Equilateral Triangle, let an equilateral triangle $\triangle DEF$ be constructed on $AB$.

From Euclid's First Postulate, let the line segment $AF$ be constructed.

Then the angle $\angle BAC$ has been bisected by the straight line segment $AF$.

Also known as

Geometric construction is also known just as construction, the geometric nature being implicit.

Sources which are careful to specify exactly where they stand may well qualify this by referring to a Euclidean construction when this is merited.

Also see

• Results about geometric constructions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): construction
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): construction