Euclid's Elements is a brilliant idea but its axioms and definitions do not meet modern standards of rigour. This project takes G. D. Birkhoff's four axioms (1932), follows the spine of the Elements, and rebuilds every theorem you meet in school from the ground up, so you can see where each theorem comes from. The full theorem atlas is now live — open any node below to watch its derivation.
Before the four axioms — what is a definition? what is an axiom? from Euclid through Hilbert to Birkhoff, why we start from here.
Points on a line are in bijection with the real numbers so that the distance between two points equals the absolute difference of their coordinates.
Any two points lie on a unique line.
Rays through a point are in bijection with [0, 2π) so that an angle equals the difference of the corresponding numbers.
If two triangles have two sides in proportion and the included angles equal, the triangles are similar.
These four are equivalent to Hilbert I + II + III + IV + V, but more direct: measurement is taken as primitive, and continuity is delegated to the real numbers.
Measurement convention: rectangle area = length × width, circle area πr², and circumference 2πr are accepted as base measure formulas — their rigorous proofs require limits and lie outside this atlas; every other area / arc-length formula is derived from these.
Definitions are not propositions waiting to be proved; they are the precise shared vocabulary of the system. Each records its dependencies so every new concept is built only from earlier axioms or definitions.
The figure formed by two rays from a common endpoint (the vertex); how wide it opens is measured by the protractor axiom into a real number between 0° and 180°, its degree measure.
Depends on · Protractor
The distance between two points is the non-negative real number that the ruler axiom measures along the segment joining them; it is 0 when the points coincide and positive otherwise, written |AB|.
Depends on · Ruler
Two lines in the same plane are parallel when they have no point in common, or coincide entirely; this is the exact form of “never meet.”
Depends on · Point–line
Two lines are perpendicular when they cross so that one of the four angles they form is a right angle; each is called a perpendicular to the other.
Depends on · Protractor
The half of a line that starts at one point and runs on without end in a single direction; that starting point is its endpoint—closed at one end, unbounded at the other.
Depends on · Point–line · Ruler
The two points on a line together with every point lying between them; the two are its endpoints, it has a definite length, and it is closed off at both ends—it does not extend.
Depends on · Point–line · Ruler
The ray from an angle's vertex that splits it into two equal halves; every point on it is equidistant from the angle's two sides.
Depends on · Angle · Ray
The curve made of all points in the plane whose distance from a fixed point (the center) equals a fixed length (the radius); the center fixes where, the radius fixes how big, and the two together pin down the circle uniquely.
Depends on · Distance
Two figures are congruent when there is a vertex correspondence under which every pair of corresponding sides is equal in length and every pair of corresponding angles is equal in measure—equivalently, one can be laid onto the other without changing shape or size, written ≅.
Depends on · Distance · Angle
The point of a segment that lies between its endpoints and is equidistant from both; it splits the segment into two equal halves, and it exists and is unique.
Depends on · Ruler · Distance
The figure enclosed by four segments joining four points in order; the four points are its vertices, the four segments its sides, and a line joining two non-adjacent vertices is a diagonal.
Depends on · Segment
Two figures are similar when there is a vertex correspondence under which every pair of corresponding angles is equal and every pair of corresponding sides is in one common ratio—same shape, possibly different size, written ∼, that shared ratio being the ratio of similarity.
Depends on · SAS similarity · Distance · Angle
The figure enclosed by joining three non-collinear points pairwise with segments; the three points are its vertices and the three segments its sides—the simplest polygon, and the most rigid shape in geometry.
Depends on · Segment · Point–line
The stretch of curve on a circle between two of its points; two points split the circle into two arcs—the shorter minor arc and the longer major arc, equal only when each is a semicircle.
Depends on · Circle
A segment joining two points on a circle; the longest chord passes through the centre and is the diameter, and every chord splits the circle into two arcs.
Depends on · Circle · Segment
A triangle with two equal sides; the equal sides are the legs and the third is the base, the angle between the legs is the apex angle and the two at the base are the base angles, which are necessarily equal.
Depends on · Triangle · Distance
A quadrilateral whose two pairs of opposite sides are each parallel; from this single condition follow equal opposite sides, equal opposite angles, and diagonals that bisect each other—the root of the special-quadrilateral family.
Depends on · Quadrilateral · Parallel
The line through a segment's midpoint that is perpendicular to it; every point on it is equidistant from the segment's two endpoints.
Depends on · Segment · Perpendicular · Midpoint
A segment from a circle's centre to any point on it—also the length of that segment; all radii of one circle are equal, and it is exactly this equality that traces the circle out.
Depends on · Circle · Segment
A triangle with one right angle (90°); the two sides forming the right angle are the legs, the longest side opposite it is the hypotenuse, and the other two angles are complementary.
Depends on · Triangle · Perpendicular
A line that meets a circle in exactly one point; that single common point is the point of tangency, and the line's distance from the centre equals the radius.
Depends on · Circle · Point–line
A quadrilateral with exactly one pair of parallel sides; the parallel sides are the bases, the other two are the legs, and when the legs are equal it is an isosceles trapezoid.
Depends on · Quadrilateral · Parallel
A parallelogram whose four angles are all right angles; from this its diagonals are equal and bisect each other, and its length and width alone determine it.
Depends on · Parallelogram · Angle
A parallelogram whose four sides are all equal; from this its two diagonals are perpendicular bisectors of each other and each bisects a pair of opposite angles.
Depends on · Parallelogram · Distance
Below is the full backbone, all the way from the axioms to the area formulae. Every incoming edge marks a logical dependency on an axiom or an earlier theorem; open any node to watch its own derivation animation.