Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun An idea that has been demonstrated as true or is assumed to be so demonstrable.
 noun Mathematics A proposition that has been or is to be proved on the basis of explicit assumptions.
from The Century Dictionary.
 To reduce to or formulate as a theorem.
 noun A universal demonstrable proposition.
 noun In geometry, a demonstrable theoretical proposition.
 noun The proposition that the velocity of a liquid flowing from a reservoir is equal to what it would have if it were to fall freely from the level in the reservoir; or, more generally, if p is the pressure, p the density, V the potential of the forces, q the resultant velocity, A a certain quantity constant along a streamline, then
 noun given by Daniel Bernoulli (1700–82) in 1738.
 noun The generalized multiplication theorem of determinants (1812).
 noun given by the eminent English mathematician George Boole (1815–64).
 noun The proposition that in the impact of inelastic bodies vis viva is always lost.
 noun The proposition that in explosions vis viva is always gained. These theorems are all due to the eminent mathematician General L. N. M. Carnot (1753–1823), who published in 1803 and and in 1786.
 noun The proposition that the ratio of the maximum mechanical effect to the whole heat expended in an expansive engine is a function solely of the two temperatures at which the heat is received and emitted: given in 1824 by Sadi Carnot (1796–1832): often called
Carnot's principle .  noun given by John Casey in 1866.
 noun The proposition that if the order of a group is divisible by a prime number, then it contains a group of the order of that prime. The extension of this—that if the order of a group is divisible by a power of a prime, it contains a group whose order is that power—is called
Cauchy and Sylow's theorem , or simply Sylow's theorem, because proved by the Norwegian L. Sylow in 1872.  noun The rule for the development of determinants according to binary products of a row and a column.
 noun The false proposition that the sum of a convergent series whose terms are all continuous functions of a variable is itself continuous.
 noun Certain other theorems are often referred to as Cauchy's, with or without further specification. All these propositions are due to the extraordinary French analyst, Baron A. L. Cauchy (1789–1857).
 noun given by E. Cesaro in 1885. It is an extension of Ceva's theorem.
 noun given by B. P. E. Clapeyron (1799–1868): otherwise called the theorem of three moments.
 noun given by L. Crocchi in 1880.
 noun given by Morgan W. Crofton in 1868. Certain symbolic expansions and a proposition in least squares are also so termed.
 noun Same as
De Moivre's property of the circle (which see, undercircle ).  noun A certain proposition in probabilities. All these are by Abraham De Moivre (1667–1754).
 noun The proposition that if two triangles ABC and A′ B′ C′ are so placed that the three straight lines through corresponding vertices meet in a point, then also the three points of intersection of corresponding sides (produced if necessary) lie in one straight line, and conversely. Both were discovered by Gérard Desargues (1593–1662).
 noun named from G. Dostor, by whom it was given in 1870. Certain corollaries from this in regard to the ellipse and hyperbola are also known as Dostor's theorems.
 noun so that in a synclastic surface
ρ _{1} andρ _{2} are the maximum and minimum radii of curvature, but in an anticlastic surface, where they have opposite signs, they are the two minima radii.  noun The proposition that in every polyhedron (but it is not true for one which enwraps the center more than once) the number of edges increased by two equals the sum of the numbers of faces and of summits.
 noun One of a variety of theorems sometimes referred to as Euler's, with or without further specification; as, the theorem that (xd/dx +
y d/dy)^{r} f(x, y) = n f(x, y); the theorem, relating to the circle, called by Euler and others Fermat's geometrical theorem; the theorem on the law of formation of the approximations to a continued fraction; the theorem of the 2, 4, 8, and 16 squares; the theorem relating to the decomposition of a number into four positive cubes. All the above (except that of Fermat) are due to Leonhard Euler (1707–831.  noun One of a number of arithmetical propositions which Fermat, owing to pressure of circumstances, could only jot down upon the margin of books or elsewhere, and the proofs of which remained unknown for the most part during two centuries, and which are still only partially understood—especially the following, called the last theorem of Fermat: the equation x +
y = z, where n is an odd prime, has no solution in integers.  noun The proposition that, if from the extremities A and B of the diameter of a circle lines AD and BE be drawn at right angles to the diameter, on the same side of it, each equal to the straight line AI or BI from A or B to the middle point of the are of the semicircle, and if through any point C in the circumference, on either side of the diameter AB, lines DCF, ECG be drawn from D and E to cut AB (produced if necessary) in F and G, then AG + BF = AB: distinguished as Fermat's geometrical theorem. This is shown in the figure by arcs from A as a center through G and from B as a center through F meeting at H on the circle.
 noun The proposition that light travels along the quickest path.
 noun given in 1820 by Sir J. F. W. Herschel (1792–1872).
 noun The proposition that forced vibrations follow the period of the exciting cause.
 noun given by the Rev. Hamnet Holditch (born 1800).
 noun The proposition that an equilibrium ellipsoid may have three unequal axes.
 noun One of a variety of other propositions relating to the transformation of Laplace's equation, to the partial determinants of an adjunct system, to infinite series whose exponents are contained in two quadratic forms, to Hamilton's equations, to distancecorrespondences for quadric surfaces, etc. All are named from their author, K. G. J. Jacobi (1804–51).
 noun The proposition that the order of a group is divisible by that of every group it contains: also called the fundamental theorem of substitutions. Both by, J. L. Lagrange (1736–1813).
 noun A proposition relating to the apparent curvature of the geocentric path of a comet. Both are named from their author, J. H. Lambert (1728–77).
 noun where the modulus of x is comprised between R and R′ : given by P. A. Laurent (1813–54).
 noun is equal to the same after development of (Du + Dv) by the binomial theorem, where D_{u} denotes differentiation as if u were constant, and D_{v} differentiation as if _{u} were constant.
 noun given by S. A. J. Lhuilier (1750–1840).
 noun a monodromic function fz can always be found having for critical points
α _{0},α _{1}, …α _{n}, etc., and such that  noun
φ n being a function for whichα n is not a critical point: given by G. MittagLeffler.  noun The proposition that the three diagonals of a quadrilateral circumscribed about a circle are all bisected by one diameter of the circle.
 noun One of the two propositions that the surface of a solid of revolution is equal to the product of the perimeter of the generating plane figure by the length of the path described by the center of gravity, and that the volume of such a solid is equal to the area of the plane figure multiplied by the same length of path. Various other theorems contained in the collection of the Greek mathematician Pappus, of the third century, are sometimes called by his name.
 noun A certain proposition concerning uniform functions connected by an algebraic relation.
 noun The proposition that a quantity of the form R = √u +
v cannot differ fromα u +β v by more than R tan ½ε whereα = cos (θ +ε )/cos ½ε ,β = sin (θ +ε )/cos ½ε ,ε = ½(Θ —θ ), tanΘ ⟩ u/v ⟩ tanθ. Both were given by General J. V. Poncelet (1788–l877).  noun The proposition that if a point be taken on each of the edges of any tetrahedron and a sphere be described through each vertex and the points assumed on the three adjacent edges, the four spheres will meet in a point: given by Samuel Roberts in 1881.
 noun where
α ,β , etc., are all the prime numbers one greater than the double of divisors of n: given in 1840 by K. G. C. von Staudt (1798–1867).  noun given by James Stirling (1696–1770).
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

Our main theorem is as follows: Once we allow for this kind of uncertainty and assume no wealth effects in preferences, the uniquely optimal social contract is laissezfaire, in which agents trade in unfettered markets with no government intervention of any kind.

Words are, after all, just empty signifiers, so what we call the theorem (the words we use to "name" the theorem) matters very little.

The factor price equalisation theorem is hard at work thanks to the fusion of insulated compartments into an open world economy.
Globalization and Inequality, Arnold Kling  EconLog  Library of Economics and Liberty

Remembering the Pythagorean theorem is a lot easier.

Ah, another point: The MM theorem is based on the fact that at times it is substantially cheaper to borrow money to put together a company from existing resources than to build those resources from scratch.
The Volokh Conspiracy » Greenspan’s ‘The Crisis’ and Modigliani and Miller

But the theorem is that 'optimum play' does exist.
Singularity Watch, Arnold Kling  EconLog  Library of Economics and Liberty

We have no graduate courses in classical mechanics and I can only assume her theorem is taught in our QFT courses.
Special Post: Noether’s First Theorem – Emmy Noether for Ada Lovelace Day

With the picture in hand, the theorem is much simpler: the radii of the blue circles sum to the radius of the orange circle.

I want to find Pythagoras and explain to him why his famous theorem is wrong.

If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach.
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