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The isoperimetric inequality bounds enclosed volume in terms of boundary size. In the plane it gives $4\pi A\le L^2$, with equality for disks; higher-dimensional forms bound the boundary area of a region of prescribed volume.

An angle subtended by a diameter of a circle is right; conversely, the hypotenuse of a right triangle inscribed in a circle is a diameter.

A rigid motion preserves Euclidean distances and therefore preserves angles, lengths, and curvatures.

An Apollonius circle is the locus of points whose distances from two fixed points have a prescribed positive ratio different from one. Ratio one gives the perpendicular bisector.

A trapezium is a quadrilateral with at least one pair of parallel opposite sides.

A plane crystallographic group is a discrete group of Euclidean plane isometries with compact quotient. Its translation subgroup is a rank-two lattice, and the quotient by translations is its finite point group. This separates the periodic translational geometry from the finite rotational and reflection symmetries.

A point group is a group of spatial symmetry operations fixing at least one common point. Choosing that point as the origin represents its operations by orthogonal matrices, so the point group is a subgroup of the orthogonal group. A pure nonzero translation cannot belong to it.^[ref]

A crystallographic point group additionally preserves a full-rank Euclidean lattice and is finite. For a crystallographic space group, the group of orthogonal linear parts is its quotient by the translation subgroup; it preserves the translation Euclidean lattice. In two dimensions the crystallographic restriction theorem allows only rotation orders $1,2,3,4,6$.

The broader definition is needed for molecular point groups. A regular planar pentagon has a fivefold rotation, so its finite point group is not crystallographic. An ideal centrosymmetric linear molecule has the infinite D infinity h molecular point group, including rotations through arbitrary angles about its axis. Thus neither finiteness nor lattice preservation belongs to the unrestricted definition.

A polygon is a closed finite chain of line segments in a plane, with successive segments meeting at vertices of a polygon. In the usual simple case, nonadjacent sides of a polygon do not intersect and the chain bounds an interior region. A triangle has three sides, a quadrilateral four, and a hexagon six. A simple $n$-sided polygon has interior angle sum $(n-2)\pi$: splitting it into $n-2$ triangles explains the formula. A regular polygon has equal side lengths and equal interior angles. Self-intersecting regular star polygons require a separate convention for their interior and signed area.

The centroid of a region $R$ with finite positive volume is the average of position with respect to uniform volume:provided the component integrals exist. For a planar region use area in place of volume; for finitely many equally weighted points use their arithmetic mean. The centroid of a uniform triangle is $(a+b+c)/3$ in terms of its vertex position vectors, the intersection of its medians of a triangle. For nonuniform mass density, weighting the same integral by density instead gives the center of mass. The vertex centroid of a general polygon need not equal the centroid of its filled interior.

A hypercube is the Cartesian product of $n$ equal-length closed intervals in Euclidean space. The centered cube $[-1,1{]}^n$ has $2^n$ vertices, each with coordinates in $\{-1,1\}$. Antipodal vertices determine $2^{n-1}$ body diagonals.

A geometric locus is the set of points satisfying a specified geometric or algebraic condition in an ambient space. For example, points at a fixed positive distance $r$ from $p$ form a circle in the plane and a sphere in three dimensions. In the complex plane, $|z-z_0|=r$ has Cartesian equation $(x-\mathrm{Re}{z}_0{)}^2+(y-\mathrm{Im}{z}_0{)}^2=r^2$, proving the circle description. A locus can also be a union, such as $x(y-1)=0$, which consists of two lines; domain restrictions in the original equation must be retained when simplifying it.

The Euclidean group $E(n)=O(n)⋉{\mathbb{R}}^n$ is the group of all isometries $x\mapsto Ax+b$ of Euclidean space, with $A$ an orthogonal matrix. Its orientation-preserving subgroup is $SE(n)=SO(n)⋉{\mathbb{R}}^n$, the semidirect product of the special orthogonal group with translations. The Euclidean motion of Euclidean three-space and the proper Euclidean motion of Euclidean three-space are the respective three-dimensional instances.

This surface consists of rays from a vertex $\mathbf{a}$ whose directions make a fixed angle $\beta$ with a unit axis vector $\hat{\mathbf{n}}$. The displayed equation follows from the dot product formula for that angle. For $0<\beta <\pi$, $\beta \ne \pi /2$, it gives one circular nappe; $\beta =\pi /2$ gives a plane, and $\beta =0,\pi$ gives a degenerate ray. Squaring the equation produces both nappes, so the unsquared sign distinguishes them. Generalized angular momentum in a radial monopole field confines particle directions to such a surface.

Figures are congruent when a rigid motion maps one onto the other, preserving all Euclidean distances.

Euclid's Elements is a thirteen-book mathematical work organized through definitions, postulates and proofs. Its topics include plane and solid Euclidean geometry, proportions and number theory. For example, Book I ends with the Pythagorean theorem and its converse, while Book IX includes a construction of even perfect numbers. Its enduring feature is the attempt to derive many geometric statements from a small explicit base of assumptions. Modern axiomatizations make additional assumptions about order, continuity and incidence explicit.^[ref]

In the Euclidean plane, precisely one line through a point outside a given line is parallel to it. This formulation is equivalent to Euclid's fifth postulate within the usual axioms of plane geometry.

A compass-and-straightedge construction uses an unmarked straightedge to draw a Euclidean line through two already constructed points, and a compass to draw a circle with a constructed centre and radius. New points arise from intersections of these lines and circles. The allowed operations matter: an accurate numerical drawing need not supply such a construction.

An ester consumed in the conductometric saponification assay, where hydrolysis under basic conditions produces ethanol and an acetate ion.

Ethyl acetoacetate is $\mathrm{C}{\mathrm{H}}_3\mathrm{COC}{\mathrm{H}}_2\mathrm{COOC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_3$, a beta-keto ester. Its central methylene group lies between two carbonyl-derived groups.

Delocalization stabilizes the corresponding enolate, making that carbon more acidic than an ordinary alkane carbon. This supports controlled carbon-carbon bond formation and subsequent transformations of the ester functionality.

For an absolutely irreducible real symmetry representation, an equivariant steady-state bifurcation with a simple transverse crossing of the scalar linear eigenvalue generically has an equilibrium branch for each axial isotropy type. Restricting the vector field to its one-dimensional fixed-point subspace reduces the existence problem to a scalar bifurcation equation. The lemma guarantees these branches; it does not exclude branches with smaller isotropy or higher-dimensional fixed-point subspaces.