* If P = orthocenter, then LMN = orthic triangle.

Triangle abc is the orthic triangle of triangle ABC

If the triangle ABC is oblique ( not right-angled ), the points of intersection of the altitudes with the sides of the triangle form another triangle, A ' B ' C ', called the orthic triangle or altitude triangle.

Also, the incenter ( that is, the center for the inscribed circle ) of the orthic triangle is the orthocenter of the original triangle.

The tangential triangle, A " B " C ", is homothetic to the orthic triangle.

The orthic triangle provides the solution to Fagnano's problem, posed in 1775, of finding for the minimum perimeter triangle inscribed in a given acute-angle triangle.

The orthic triangle of an acute triangle gives a triangular light route.

Trilinear coordinates for the vertices of the orthic triangle are given by

The joining of these three orthogonal points into a triangle generates an orthic triangle that is common to all the four possible triangles formed from the four orthocentric points taken three at a time.

Note that the incenter of this common orthic triangle must be one of the original four orthocentric points.

Furthermore, the three remaining points become the excenters of this common orthic triangle.

The orthocentric point that becomes the incenter of the orthic triangle is that orthocentric point closest to the common nine-point center.

It is normal to choose the orthocentric point that is the incenter of the orthic triangle as H the orthocenter of the outer three orthocentric points that are chosen as a reference triangle ABC.

The orthic axis associated with a normalized orthocentric system A, B, C and H, where ABC is the reference triangle, is a line that passes through three intersection points formed when each side of the orthic triangle meets each side of the reference triangle.

So if a rectangular hyperbola is drawn through four orthocentric points it will have one fixed center on the common nine-point circle but it will have four perspectors one on each of the orthic axes of the four possible triangles.

In a normalized orthocentric system the orthic inconic that is tangent to the sides of the triangle ABC is an inellipse and the orthic inconics of the other three possible triangles are hyperbolas.

The circumcubic known as the orthocubic-K006 is interesting in that it passes through three orthocentric systems as well as the three vertices of the orthic triangle ( but not the orthocenter of the orthic triangle ).

The seven intersections referred above are A, B, C, H ( the original orthocentric points ) and H < sub > A </ sub >, H < sub > B </ sub >, H < sub > C </ sub > ( the feet of the altitudes of triangle ABC and the vertices of the orthic triangle ).

The contacts of these inconics with the four possible triangles occur at the vertices of their common orthic triangle.

This relationship between the orthic triangle and the original four orthocentric points leads directly to the fact that the incenter and excenters of a reference triangle form an orthocentric system.

They each have their own orthic axis.

The four Euler lines of an orthocentric system are orthogonal to the four orthic axes of an orthocentric system.

Note that the locus of the perspectors of this family of rectangular hyperbolas will always lie on the four orthic axes.

The four possible triangles have a set of four inconics known as the orthic inconics that share certain properties.

These four orthic inconics also share the same Brianchon point, H, the orthocentric point closest to the common nine-point center.

The centers of these orthic inconics are the symmedian points, K of the four possible triangles.

An American in Paris is scored for 3 flutes ( 3rd doubling on piccolo ), 2 oboes, English horn, 2 clarinets in B flat, bass clarinet in B flat, 2 bassoons, 4 horns in F, 3 trumpets in B flat, 3 trombones, tuba, timpani, snare drum, bass drum, triangle, wood block, cymbals, low and high tom-toms, xylophone, glockenspiel, celesta, 4 taxi horns resembling the pitches A, B, C and D, alto saxophone / soprano saxophone, tenor saxophone / soprano saxophone / alto saxophone, baritone saxophone / soprano saxophone / alto saxophone, and strings.

If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

There, a small peninsular area is still called " Angeln " today and is formed as a triangle drawn roughly from modern Flensburg on the Flensburger Fjord to the City of Schleswig and then to Maasholm, on the Schlei inlet.

The northern triangle of Times Square is technically Duffy Square, dedicated in 1937 to Chaplain Francis P. Duffy of New York City's " Fighting 69th " Infantry Regiment ; a memorial to Duffy is located there, along with a statue of George M. Cohan.

* In a diode, it is the positive terminal at the tail of the arrow symbol ( base of the triangle ), where current flows into the device.

In reality, this " triangle arbitrage " is so simple that it almost never occurs.

At the intersection with 72nd street, the triangle of tiny Verdi Square is surrounded by several notable apartment buildings, including The Ansonia, and the Florentine palazzo occupied by Apple Bank for Savings.

Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.

The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.

: This is the third line in the Pascal: s triangle.

The goat's head is formed by the triangle of stars ι Cap, θ Cap, and ζ Cap.

* Other: Correspondences include blood, the guitar, thunder, drum beat rubies and in writing fire is sometimes represented by a red upwards triangle.

The alchemical symbol for air is an upward-pointing triangle, bisected by a horizontal line.

It is sometimes represented by its Tattva or by a downward pointing triangle with a horizontal line through it.

It is based on the semaphore symbols for " N " ( two flags held 45 degrees down on both sides, forming the triangle at the bottom ) and " D " ( two flags, one above the head and one at the feet, forming the vertical line ) ( for Nuclear Disarmament ) within a circle.

A river delta ( originally, the Nile River delta ) is so named because its shape approximates the upper-case letter delta ( the shape is a triangle ).

In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulation DT ( P ) such that no point in P is inside the circumcircle of any triangle in DT ( P ).

As mentioned above, if a triangle is non-Delaunay, we can flip one of its edges.

This leads to a straightforward algorithm: construct any triangulation of the points, and then flip edges until no triangle is non-Delaunay.

When a vertex v is added, we split in three the triangle that contains v, then we apply the flip algorithm.

A string tied at each end to the two pins and the tip of a pen is used to pull the loop taut so as to form a triangle.