Geometric Construction/Examples

Examples of Geometric Constructions

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 $AF$.