Euclidean Geometry Revisited - Examples and Constructions

Classical Euclidean geometry emphasizes constructions using only a straightedge (an unmarked ruler) and compass. These tools correspond to the first three postulates of Euclid: drawing a straight line between two points, extending a line segment, and drawing a circle with a given center and radius.

ExampleBisecting an Angle

To bisect an angle $\angle BAC$ :

Draw a circle centered at $A$ intersecting rays $AB$ and $AC$ at points $D$ and $E$
Draw circles of equal radius centered at $D$ and $E$
Let $F$ be an intersection point of these circles
The ray $AF$ bisects $\angle BAC$

This construction guarantees that $\angle BAF = \angle FAC$ by the SSS congruence criterion applied to triangles $\triangle ADF$ and $\triangle AEF$ .

The question of which geometric constructions are possible with compass and straightedge has deep connections to field theory and Galois theory. A length is constructible if and only if it belongs to a field extension of $\mathbb{Q}$ obtained by a finite sequence of quadratic extensions. This algebraic criterion resolves ancient problems.

ExampleImpossible Constructions

Three famous problems from antiquity were proven impossible using only compass and straightedge:

Doubling the cube: Constructing a cube with volume $2V$ given a cube of volume $V$ . This requires constructing $\sqrt[3]{2}$ , which generates a cubic extension of $\mathbb{Q}$ .
Trisecting an angle: Dividing an arbitrary angle into three equal parts. While specific angles (like $90°$ ) can be trisected, the general problem requires solving cubic equations.
Squaring the circle: Constructing a square with the same area as a given circle. Since $\pi$ is transcendental (Lindemann, 1882), this is impossible.

The construction of regular polygons reveals beautiful number theory. Gauss proved in 1796 that a regular $n$ -gon is constructible if and only if $n$ is the product of a power of 2 and distinct Fermat primes (primes of the form $2^{2^k} + 1$ ). Known Fermat primes are $F_0=3$ , $F_1=5$ , $F_2=17$ , $F_3=257$ , and $F_4=65537$ . Thus regular 17-gons and 257-gons are constructible.

DefinitionConstructible Numbers

A real number $\alpha$ is constructible if, starting from points at distance 1, we can construct a line segment of length $|\alpha|$ using compass and straightedge. The set of constructible numbers forms a field containing all numbers obtainable from $\mathbb{Q}$ by taking square roots finitely many times.

Circle geometry provides rich examples. The nine-point circle of a triangle passes through nine significant points: the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from vertices to the orthocenter. The Euler line connects the centroid, circumcenter, and orthocenter, with the centroid dividing the segment in a 2:1 ratio.

Remark

The study of triangle centers has identified thousands of special points, each defined by specific geometric properties. The ETC (Encyclopedia of Triangle Centers) catalogs these systematically. Classical centers include the centroid (center of mass), circumcenter (center of circumscribed circle), incenter (center of inscribed circle), and orthocenter (intersection of altitudes).

Inversion in a circle provides a powerful transformation technique. Given a circle of radius $r$ centered at $O$ , the inverse of a point $P$ is the point $P'$ on ray $OP$ such that $|OP| \cdot |OP'| = r^2$ . This transformation maps circles and lines to circles and lines (with lines through $O$ mapping to themselves), providing elegant solutions to many geometric problems. It also reveals the conformal structure underlying Euclidean geometry.