Geometric Constructions: Compass & Straightedge

Euclid's Geometry: The Axiomatic System & 5 Postulates

Euclid's geometry is the plane geometry built in Euclid's Elements from a small set of definitions, five postulates, and common notions, with every later result proved by logic. This article gives an overview of that axiomatic system, states the five postulates in brief, and shows how the fifth led to non-Euclidean geometry.

Euclid's Fifth Postulate: The Parallel Postulate

Euclid's fifth postulate - the parallel postulate - says that if a line crossing two lines makes the interior angles on one side add to less than 180°, those two lines meet on that side. Its simpler equivalent (Playfair's axiom) is that through a point not on a line, exactly one parallel can be drawn. This article states the postulate precisely, explains why 2,000 years of attempted proofs failed, and shows how denying it created non-Euclidean geometry.

Euclid's Axioms and Postulates: The 7 Axioms and 5 Postulates

Euclid's axioms and postulates are the twelve self-evident assumptions on which all of classical geometry is built. The 7 axioms (Euclid's "common notions") are general truths about equality that apply across mathematics; the 5 postulates are assumptions specific to geometry, such as "a straight line can be drawn between any two points." This article lists all twelve, explains each, and clarifies how an axiom differs from a postulate.

Equation of a Plane: General, Normal & Intercept Forms

The equation of a plane in 3D space is $ax + by + cz + d = 0$, where $(a, b, c)$ is a vector normal (perpendicular) to the plane. This article covers the general, normal, point-normal, three-point, and intercept forms, derives each from a normal vector, works six examples including finding a plane through three points, and clears up the sign and normal-vector mistakes students make most.

Eccentricity of Parabola: Why It Equals 1

The eccentricity of a parabola is exactly $e = 1$, because every point on a parabola is the same distance from the focus as from the directrix. This article defines eccentricity, derives why a parabola gives $e = 1$, compares it with the other conics, and works through six examples.

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