What Geometry Studies
Geometry is the study of shape, size and position. Plane geometry works in a flat surface with a small set of objects — points, lines, angles and the figures they enclose — and asks which of their properties must always hold, whatever the particular drawing looks like.
What sets geometry apart from measurement is proof. Measuring the angles of one triangle tells you about that triangle; a proof tells you about every triangle at once. Each result is derived from a few accepted starting points, the axioms, and from results proved earlier, so the whole subject is one connected chain of reasoning.
This section starts from those proofs. Each figure gets an interactive visual tool where the argument unfolds one step at a time, with the rule behind every step named beside it. You can drag the figure at any point and watch the conclusion survive the change — the clearest way to see why a proof is general.
Sections
Visual tools are organized by figure. Each figure has its own player, and each theorem about that figure is a separate scenario: a sequence of scenes that adds constructions and markings, highlights what the current step uses, and keeps a running log of the argument with its justification.
The figures are the shapes the proofs are about. Triangles come first because most of plane geometry rests on them: congruence and similarity are defined for triangles, and results about circles, quadrilaterals and polygons are usually proved by cutting those shapes into triangles.
Triangle
The simplest polygon and the base of most proofs in the plane: angle relationships, isosceles and equilateral triangles, congruence and the Pythagorean theorem, each on a triangle you can reshape.
- Triangle angle sum
Circle
All points at one distance from a center. Inscribed and central angles, Thales' theorem, tangents meeting radii at right angles, chords and arcs.
Quadrilateral
Four-sided figures and their families: parallelograms, rectangles, rhombi, trapezoids. Properties of sides and diagonals, and the 360° angle sum.
Polygon
Figures with any number of sides. Interior angle sums of (n − 2) · 180°, exterior angles that always total 360°, and regular polygons.
Core Ideas
Points, Lines and Angles
The basic objects. Two points determine a line; two rays from a common point form an angle, measured in degrees.
Parallel Lines
Lines that never meet. A transversal crossing them creates pairs of equal angles — the tool behind many angle proofs.
Congruence
Figures with the same shape and size. For triangles, three matching parts are enough: SSS, SAS, ASA.
Similarity
Same shape, different size. Corresponding angles are equal and corresponding sides share one ratio.
Area
The space a figure covers. Many area formulas are proved by cutting a figure apart and rearranging the pieces.
How a Proof Works Here
- GivenThe figure and what is known about it, marked on the canvas.
- ConstructAuxiliary lines or points that the argument needs, added one at a time.
- JustifyEach step names its rule — a postulate, a definition or an earlier theorem.
- ConcludeThe statement follows. Drag the figure to see it hold for every shape.