A more recent proof demonstrates both of these impossibilities by proving still more broadly that a Pythagorean triangle cannot have two unspecified sides each of which is each either a square or twice a square, since there is no smallest such triangle.

In the triangle between the cities of Amsterdam, Haarlem and Leiden the Haarlemmermeer polder demonstrates an extraordinary step in the scale of land drainage.

The Reuleaux triangle can be rotated whilst always touching each side of the square in a single point ; this demonstrates that its width ( separation between parallel supporting lines ) is constant in all directions.

Macewen demonstrates his triangle to Lane, Hirschfelder, Barkan and Stillman.

" Homer parodies the scene in the 1939 film Wizard of Oz when Scarecrow demonstrates his newly acquired intelligence by reciting the law that governs the lengths of the sides of an isosceles triangle.

The town was built on the southern declivity, and was surrounded by a wall, which, allowing for the natural irregularities of the soil, represented a triangle, with the castle at the summit — a form observable in many of the ancient cities of Greece.

However, Pascal developed many uses of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique ( 1653 ).

After the camps were liberated at the end of the Second World War, many of the pink triangle prisoners were often simply re-imprisoned by the Allied-established Federal Republic of Germany.

Miyamoto came up with many characters and plot concepts, but he eventually settled on a gorilla / carpenter / girlfriend love triangle that mirrored the rivalry between Bluto and Popeye for Olive Oyl.

In many configurations, that triangle is about half a meter wide, making its point extremely close to the mile start line, which would be slightly less than two feet from the marked relay zone ( the widest part of the triangle, or line ).

The triangle of land now includes many homes around Old Brownsboro Court, and a small shopping center with a Rite Aid.

On many men, the pubic patch tapers upwards to a line of hair pointing towards the navel ( see abdominal hair ), roughly a more upward-pointing triangle.

This is different from many 3D computer modeling packages, which typically use triangle meshes to compose all objects.

There are many documented cubics that pass through a reference triangle and its orthocenter.

After many failed attempts to derive the parallel postulate from other axioms, the study of the still hypothetical hyperbolic geometry by Johann Heinrich Lambert ( 1728 – 1777 ) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle ( where the sum of angles is less than 180 °).

Thus, in spherical geometry angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects ; for example, the sum of the interior angles of a triangle exceeds 180 degrees.

Through the confession, and the staged scenes acted out by the male prisoners in the prison chapel, we learn that the bishop and the prisoner were part of a gay love triangle, and that the bishop was, in fact, responsible for the death of a young man many years ago.

Constructions for the regular triangle, square, pentagon, and polygons with 2 < sup > h </ sup > times as many sides had been given by Euclid, but constructions based on the Fermat primes other than 3 and 5 were unknown to the ancients.

" Although the romantic love triangle is formally identical to the friendship triad, as many have noted their actual implications are quite different .... Romantic love is typically viewed as an exclusive relationship, whereas friendship is not.

The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus ( dialogue ) 54d and by many other ancient authors.

In the end, Rice wasn't able to win the hearts of Los Angeles fans after being traded for fan-favorite, Eddie Jones with many citing suspect defense and Rice's inability to perform in the triangle offense.

While some strangles can be done by lifting the other person in the air, off their toes, pinned against a wall, or naturally lifted off the ground which requires physical strength, many technical strangles such as Rear naked choke ( hadaka-jime ) or triangle choke ( sankaku-jime ) requires much less physical strength and can be executed by a smaller person on a larger assailant.

New England also benefited from the trade, as many merchants from New England, especially the state of Rhode Island, replaced the role of Europe in the triangle.

There are many examples, including the one known as the Maindee triangle in Newport, South Wales.

This showed that the triangle of land at the northwestern point of Eastern Rhodesia from Pweto to as far south as the Lunchinda River was under Northern Rhodesia even though the Belgian Congo had administered it for many years.

* The triangle groups include 3 finite groups ( the spherical ones, corresponding to sphere ), 3 groups of quadratic growth ( the Euclidean ones, corresponding to Euclidean plane ), and infinitely many groups of exponential growth ( the hyperbolic ones, corresponding to the hyperbolic plane ).

Some mallets, such as a triangle wand, are normally used only with a specific instrument, while others are used on many different instruments.

* Vladimir Nabokov's novel Ada tells the story of an affair between siblings who believe they are merely cousins ( relations between first cousins being acceptable and even common in many eras ), including a love triangle with the eponymous character's believed full, actually half sister.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

When a mathematician reasons from the statement " ABC is a triangle " to the statement " triangles exist ", she is not asserting that triangles are made up of atoms but rather that triangles exist within a particular mathematical model.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.

Although Escher did not have mathematical training — his understanding of mathematics was largely visual and intuitive — Escher's work had a strong mathematical component, and more than a few of the worlds which he drew are built around impossible objects such as the Necker cube and the Penrose triangle.

Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients ; these particular coefficients are tabulated in Pascal's triangle.

The neutral point in a three phase system exists at the mathematical center of an equilateral triangle formed by the three phase points, and the phase-to-phase voltage is accordingly times the phase-to-neutral voltage.

His correspondence with Fermat contains the only extant mathematical proof by Fermat, a demonstration, using his method of descent, that the area of a Pythagorean triangle cannot be a square.

But, no mathematical re-arrangement of a triangle could allow its three internal angles to sum to anything but 180 degrees.

The formula can by proved using mathematical induction and starting with a triangle for which the angle sum is 180 °, and then adding a vertex and two sides, etc.

* Sir Roger, born 1931, mathematical physicist ( with whom Lionel co-authored papers on the Penrose triangle )

* Kepler triangle # A mathematical coincidence

Two popular filter metrics for classification problems are correlation and mutual information, although neither are true metrics or ' distance measures ' in the mathematical sense, since they fail to obey the triangle inequality and thus do not compute any actual ' distance ' – they should rather be regarded as ' scores '.

The simpler polygon formed by removing this triangle has a 3-coloring by mathematical induction, and this coloring is easily extended to the one additional vertex of the removed triangle.