For equal masses and relative position $\mathbf{r}$, the individual positions are ${\mathbf{r}}_A=\mathbf{R}-\mathbf{r}/2$ and ${\mathbf{r}}_B=\mathbf{R}+\mathbf{r}/2$, where the center of mass has uniform velocity in an isolated system. Bound inverse-square force motion makes $\mathbf{r}$ an ellipse; each mass follows a half-size ellipse in the center-of-mass frame. Adding a nonzero center-of-mass drift generally makes their laboratory trajectories nonclosed. The relative orbital period is not necessarily a period of either laboratory position.

For two particles of equal mass $m$ and opposite charges $\pm q$ separated by $s$, the centre of mass lies halfway between them. In a nonradiating classical model, each radius is $s/2$ and $k_e{q}^2/s^2=m{\omega }^{2}s/2$. Hence ${\omega }^2=2k_e{q}^2/(m{s}^3)$ and $T=2\pi \sqrt{m{s}^3/(2k_e{q}^2)}$. Equivalently, relative motion has reduced mass $m/2$. Radiation and quantum binding invalidate an indefinitely stable classical orbit for actual microscopic charges.

Net daylight ozone production occurs when peroxy radicals oxidize NO to NO2 without using ozone, followed by NO2 photodissociation and oxygen-atom addition to O2. The competing reaction NO + O3 to NO2 + O2 only recycles odd oxygen. Radical initiation, termination, precursor loading and transport determine the net rate.

Local ozone consumption by the reaction $\mathrm{NO}+{\mathrm{O}}_3\to \mathrm{N}{\mathrm{O}}_2+{\mathrm{O}}_2$. Reducing fresh NO emissions can raise local measured ozone even while decreasing regional ozone production. This distinction matters when interpreting precursor-control experiments and observations.

An isosceles triangle has two equal sides. Its angles opposite those sides are equal: reflecting the triangle across the bisector of the angle between the equal sides exchanges them and fixes the opposite side as a set. A triangle formed by two radii of one circle is therefore isosceles.

The common intersection of a triangle's three altitude lines. Each line passes through a vertex and is orthogonal to the opposite side. The intersection can lie inside or outside the triangle, or at its right-angle vertex. An orthogonal projection onto an intercept plane supplies a concrete three-dimensional construction of one.

In a nondegenerate triangle, let $a,b,c$ be the side lengths opposite the interior angles $A,B,C$. Its area $K$ satisfies $2K=bc\sin A=ca\sin B=ab\sin C$. Division by $abc>0$ proves the law of sines. The area formula itself follows by taking the magnitude of the cross product of two edge vectors. Unlike the law of cosines, this relation alone may leave an ambiguity between two supplementary angles when two sides and a nonincluded angle are given.

A Fermat point of a triangle minimizes the sum of Euclidean distances to its three vertices. If every interior angle is less than $120^{\circ }$, the minimizer lies inside, and the three joining segments meet at pairwise angles of $120^{\circ }$. Indeed the gradient of the sum at an interior minimizer is the sum of three unit vectors toward the vertices; three unit vectors sum to zero exactly in this configuration. Convexity of distance makes this stationary point a global minimizer. If one interior angle is at least $120^{\circ }$, that vertex is a minimizer instead. For an equilateral triangle, the Fermat point is its centroid.

An internal angle bisector of a triangle divides the opposite side in the ratio of the two adjacent sides. If $AD$ bisects the angle at $A$ in triangle $ABC$, then $BD/DC=AB/AC$. Comparing the areas of triangles $ABD$ and $ACD$ proves this: their shared altitude gives the ratio $BD/DC$, while their equal included angles give $AB/AC$.

The intersection of a triangle's internal angle bisectors. It is the center of the circle tangent to all three sides, and its distance to any side is the inradius.

The radius of a triangle's circumscribed circle, whose center is the circumcenter. For side lengths $a,b,c$ and area $A$, it satisfies $R=abc/(4A)$.

For a triangle with side lengths $a,b,c$ and semiperimeter $s=(a+b+c)/2$, its area is $A=\sqrt{s(s-a)(s-b)(s-c)}$. This expresses the area using only the three side lengths.

A median joins a vertex of a polygon to the midpoint of the opposite side of a triangle. For side lengths $a,b,c$, the median to the side of length $c$ has length $m_c=\frac{1}{2}\sqrt{2a^2+2b^2-c^2}$, by Apollonius's theorem.

For a triangle with vertices $A,B,C$ and midpoint $M$ of $BC$, Apollonius's theorem statesThus the length of a median of a triangle is determined by the three side lengths. Put the origin at $M$, so $B=v$, $C=-v$, $A=u$. Expanding the dot products in $|u-v{|}^2+|u+v{|}^2$ cancels the cross terms, giving $2|u{|}^2+2|v{|}^2$. Since $|BC|=2|v|$, this proves the formula and relates it to the parallelogram law. For sides $AB=AC=5$, $BC=6$, the median has length four.

Insufficient availability of one or more amino acids relative to cellular requirements. Its translational consequences include substrate limitation and regulated changes in initiation and stress adaptation.

Processes recognizing problematic stalled ribosomes and managing their incomplete polypeptides and associated components.

A eukaryotic initiation factor carrying initiating methionyl-tRNA in a GTP-bound ternary complex. Its alpha-subunit phosphorylation can inhibit nucleotide exchange by eIF2B and lower initiation capacity.

The nucleotide-exchange factor that promotes replacement of GDP by GTP on eIF2. Inhibition of this exchange reduces the supply of initiation-competent ternary complex.

A translated reading region upstream of a main coding region in a messenger RNA. Its effects depend on initiation, termination, scanning and reinitiation, rather than its mere presence.