Geometry Glossary
The vocabulary of contest geometry, grouped so the definitions sit next to their relatives.
Basic Objects
Point — a position with no size. Undefined, taken as given.
Line — extends without end in both directions, no thickness. Undefined.
Plane — a flat surface extending without end. Undefined.
Segment — the part of a line between two endpoints.
Ray — starts at a point and extends without end in one direction.
Collinear — points lying on a single line. Coplanar — points lying in a single plane.
Angle — the figure formed by two rays sharing an endpoint (the vertex).
Axiom / postulate — a statement assumed true without proof, from which everything else follows.
Theorem — a statement proved from axioms and earlier theorems.
Angles
Acute — less than 90°. Right — exactly 90°. Obtuse — between 90° and 180°. Straight — exactly 180°. Reflex — between 180° and 360°.
Complementary — two angles summing to 90°. Supplementary — summing to 180°.
Vertical angles — the opposite pair formed by two crossing lines. Always equal.
Transversal — a line crossing two or more others.
Corresponding, alternate, and co-interior angles — the angle pairs formed when a transversal crosses two parallel lines. Corresponding and alternate pairs are equal; co-interior pairs are supplementary. This is what makes angle chasing possible.
Angle chasing — working systematically through a figure labelling angles until the required one is determined. The most-used technique in contest geometry.
Triangles
Scalene / isosceles / equilateral — no equal sides / two equal sides / three equal sides.
Acute / right / obtuse triangle — classified by its largest angle.
Hypotenuse — the side opposite the right angle, always the longest.
Altitude — a perpendicular segment from a vertex to the opposite side (or its extension).
Median — a segment from a vertex to the midpoint of the opposite side.
Angle bisector — a ray or segment cutting an angle exactly in half.
Perpendicular bisector — a line through a segment's midpoint at right angles to it.
Midsegment — the segment joining two sides' midpoints. Parallel to the third side and half its length.
Centroid — where the three medians meet. Divides each median in a 2:1 ratio, and is the triangle's centre of mass.
Orthocentre — where the three altitudes meet.
Incentre — where the three angle bisectors meet. The centre of the inscribed circle.
Circumcentre — where the three perpendicular bisectors meet. The centre of the circumscribed circle.
Triangle inequality — any two sides sum to more than the third.
Angle bisector theorem — an angle bisector divides the opposite side in the ratio of the adjacent sides.
Heron's formula — the area of a triangle from its three side lengths, via the semi-perimeter.
Congruence and Similarity
Congruent — identical in shape and size. Written ≅.
Similar — identical in shape, possibly different in size. Written ~.
SSS, SAS, ASA, AAS, HL — the criteria establishing triangle congruence.
AA, SAS, SSS similarity — the criteria establishing triangle similarity.
Scale factor — the ratio between corresponding lengths in similar figures. Areas scale by its square, volumes by its cube.
Circles
Radius — centre to circumference. Diameter — across the circle through the centre; twice the radius.
Chord — a segment joining two points on the circle.
Arc — a portion of the circumference. Sector — the region between two radii and an arc. Segment — the region between a chord and an arc.
Tangent — a line touching the circle at exactly one point. Always perpendicular to the radius at that point.
Secant — a line crossing the circle at two points.
Central angle — vertex at the centre. Inscribed angle — vertex on the circumference.
Inscribed angle theorem — an inscribed angle is half the central angle subtending the same arc. The source of most circle results.
Cyclic quadrilateral — four points lying on one circle. Its opposite angles are supplementary — and the converse is true, which is how you prove four points concyclic.
Concyclic — lying on a common circle.
Power of a point — for a point and a circle, the constant product of the distances along any line through the point to the circle's two intersections. Covers intersecting chords, secants, and tangents in one statement.
Incircle / circumcircle — the circle inscribed in a polygon / passing through all its vertices.
Apothem — the distance from a regular polygon's centre to the midpoint of a side.
Polygons
Polygon — a closed figure made of straight segments.
Regular polygon — all sides and all angles equal.
Convex — every interior angle under 180°. Concave — at least one over.
Interior angle sum — (n − 2) × 180° for an n-sided polygon. Exterior angles always total 360°.
Parallelogram — both pairs of opposite sides parallel. Diagonals bisect each other.
Rhombus — a parallelogram with four equal sides. Diagonals are perpendicular.
Trapezoid — exactly one pair of parallel sides.
Kite — two distinct pairs of adjacent equal sides.
Tessellation — a tiling of the plane with no gaps or overlaps.
Solids
Polyhedron — a solid with flat polygonal faces.
Prism — two congruent parallel bases joined by parallelograms. Pyramid — a base and an apex.
Cylinder, cone, sphere — the curved-surface solids.
Net — a flat pattern that folds into a solid.
Cross-section — the plane figure produced by slicing a solid.
Lateral surface area — the area of the sides, excluding the bases.
Coordinates and Transformations
Distance formula — the Pythagorean theorem in coordinates.
Midpoint formula — the average of the endpoints' coordinates.
Slope — rise over run. Parallel lines share a slope; perpendicular slopes multiply to −1.
Shoelace formula — a polygon's area directly from its vertices' coordinates.
Translation — a slide. Rotation — a turn about a point. Reflection — a flip across a line. Dilation — a resize about a point.
Rigid motion (isometry) — a transformation preserving distance: translation, rotation, or reflection. Congruence is exactly "related by a rigid motion".
Line of symmetry — a line across which a figure reflects onto itself.
Trigonometry
Sine, cosine, tangent — the ratios relating an acute angle to the sides of a right triangle.
Law of sines — a/sin A = b/sin B = c/sin C, for any triangle.
Law of cosines — c² = a² + b² − 2ab·cos C. The Pythagorean theorem with a correction term for non-right angles.
Reasoning
Proof — a complete argument establishing a claim in every case.
Converse — the statement with hypothesis and conclusion swapped. Not automatically true.
Auxiliary line — a line added to a figure to expose a useful configuration. Choosing the right one is frequently the whole problem.
Without loss of generality (WLOG) — marks an assumption that costs nothing because remaining cases are symmetric.
Counterexample — one case that disproves a general claim.
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