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54-404: The celestial poles are also the poles of the celestial equatorial coordinate system , meaning they have declinations of +90 degrees and −90 degrees (for the north and south celestial poles, respectively). Despite their apparently fixed positions, the celestial poles in the long term do not actually remain permanently fixed against the background of the stars. Because of a phenomenon known as

108-548: A right-handed convention. The origin at the centre of Earth means the coordinates are geocentric , that is, as seen from the centre of Earth as if it were transparent . The fundamental plane and the primary direction mean that the coordinate system, while aligned with Earth's equator and pole , does not rotate with the Earth, but remains relatively fixed against the background stars . A right-handed convention means that coordinates increase northward from and eastward around

162-415: A circle with a certain radius, placing the point of the compass on the circle, and drawing another circle with the same radius; the two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and the points of intersection. An alternative way to construct an equilateral triangle is by using Fermat prime . A Fermat prime is a prime number of

216-514: A family of polyhedra incorporating a band of alternating triangles. When the antiprism is uniform , its bases are regular and all triangular faces are equilateral. As a generalization, the equilateral triangle belongs to the infinite family of n {\displaystyle n} - simplexes , with n = 2 {\displaystyle n=2} . Equilateral triangles have frequently appeared in man-made constructions and in popular culture. In architecture, an example can be seen in

270-437: A left-handed system, measures the angular distance of an object westward along the celestial equator from the observer's meridian to the hour circle passing through the object. Unlike right ascension, hour angle is always increasing with the rotation of Earth . Hour angle may be considered a means of measuring the time since upper culmination , the moment when an object contacts the meridian overhead. A culminating star on

324-526: A plane, known as the trigonal planar molecular geometry . In the Thomson problem , concerning the minimum-energy configuration of n {\displaystyle n} charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the best solution known for n = 3 {\displaystyle n=3} places

378-485: A point P {\displaystyle P} in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when P {\displaystyle P} is the centroid. In no other triangle is there a point for which this ratio is as small as 2. This is the Erdős–Mordell inequality ;

432-514: A result of the method of measuring right ascensions by timing the passage of objects across the meridian as the Earth rotates . There are ⁠ 360° / 24 ⁠ = 15° in one hour of right ascension, and 24 of right ascension around the entire celestial equator . When used together, right ascension and declination are usually abbreviated RA/Dec. Alternatively to right ascension , hour angle (abbreviated HA or LHA, local hour angle ),

486-442: A small oscillation of the Earth's axis, nutation . In order to fix the exact primary direction, these motions necessitate the specification of the equinox of a particular date, known as an epoch , when giving a position. The three most commonly used are: A position in the equatorial coordinate system is thus typically specified true equinox and equator of date , mean equinox and equator of J2000.0 , or similar. Note that there

540-411: A star's position differs from observer to observer based on their positions on the Earth's surface, and is continuously changing with the Earth's rotation. Telescopes equipped with equatorial mounts and setting circles employ the equatorial coordinate system to find objects. Setting circles in conjunction with a star chart or ephemeris allow the telescope to be easily pointed at known objects on

594-435: A straight line and place the point of the compass on one end of the line, then swing an arc from that point to the other point of the line segment; repeat with the other side of the line, which connects the point where the two arcs intersect with each end of the line segment in the aftermath. If three equilateral triangles are constructed on the sides of an arbitrary triangle, either all outward or inward, by Napoleon's theorem

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648-467: A stronger variant of it is Barrow's inequality , which replaces the perpendicular distances to the sides with the distances from P {\displaystyle P} to the points where the angle bisectors of ∠ A P B {\displaystyle \angle APB} , ∠ B P C {\displaystyle \angle BPC} , and ∠ C P A {\displaystyle \angle CPA} cross

702-459: Is a celestial coordinate system widely used to specify the positions of celestial objects . It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth , a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere (forming the celestial equator ), a primary direction towards the March equinox , and

756-413: Is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon , occasionally known as the regular triangle . It is the special case of an isosceles triangle by modern definition, creating more special properties. The equilateral triangle can be found in various tilings , and in polyhedrons such as

810-479: Is always (nearly) equal to the observer's geographic latitude (though it can, of course, only be seen from locations in the Northern Hemisphere). Polaris is near the north celestial pole for only a small fraction of the 25,700-year precession cycle. It will remain a good approximation for about 1,000 years, by which time the pole will have moved closer to Alrai ( Gamma Cephei ). In about 5,500 years,

864-543: Is an arbitrary point in the plane of an equilateral triangle A B C {\displaystyle ABC} but not on its circumcircle , then there exists a triangle with sides of lengths P A {\displaystyle PA} , P B {\displaystyle PB} , and P C {\displaystyle PC} . That is, P A {\displaystyle PA} , P B {\displaystyle PB} , and P C {\displaystyle PC} satisfy

918-524: Is analogous to terrestrial latitude . The right ascension symbol α , (lower case "alpha", abbreviated RA) measures the angular distance of an object eastward along the celestial equator from the March equinox to the hour circle passing through the object. The March equinox point is one of the two points where the ecliptic intersects the celestial equator. Right ascension is usually measured in sidereal hours, minutes and seconds instead of degrees,

972-520: Is called a deltahedron . There are eight strictly convex deltahedra: three of the five Platonic solids ( regular tetrahedron , regular octahedron , and regular icosahedron ) and five of the 92 Johnson solids ( triangular bipyramid , pentagonal bipyramid , snub disphenoid , triaugmented triangular prism , and gyroelongated square bipyramid ). More generally, all Johnson solids have equilateral triangles among their faces, though most also have other other regular polygons . The antiprisms are

1026-520: Is formulated as t 2 = R ( R − 2 r ) {\displaystyle t^{2}=R(R-2r)} . As a corollary of this, the equilateral triangle has the smallest ratio of the circumradius R {\displaystyle R} to the inradius r {\displaystyle r} of any triangle. That is: R ≥ 2 r . {\displaystyle R\geq 2r.} Pompeiu's theorem states that, if P {\displaystyle P}

1080-433: Is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations. A star 's spherical coordinates are often expressed as a pair, right ascension and declination , without a distance coordinate. The direction of sufficiently distant objects is the same for all observers, and it is convenient to specify this direction with the same coordinates for all. In contrast, in the horizontal coordinate system ,

1134-535: Is removed to the centre of the Sun . It is commonly used in planetary orbit calculation. The three astronomical rectangular coordinate systems are related by ξ = x + X η = y + Y ζ = z + Z {\displaystyle {\begin{aligned}\xi &=x+X\\\eta &=y+Y\\\zeta &=z+Z\end{aligned}}} Equilateral triangle An equilateral triangle

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1188-470: The Sierpiński triangle (a fractal shape constructed from an equilateral triangle by subdividing recursively into smaller equilateral triangles) and Reuleaux triangle (a curved triangle with constant width , constructed from an equilateral triangle by rounding each of its sides). Equilateral triangles may also form a polyhedron in three dimensions. A polyhedron whose faces are all equilateral triangles

1242-739: The Southern Sky . These are marked in astronomy books as the Large and Small Magellanic Clouds (the LMC and the SMC). These "clouds" are actually dwarf galaxies near the Milky Way . Make an equilateral triangle, the third point of which is the south celestial pole. Like before, the SMC, LMC, and the pole will all be points on an equilateral triangle on an imaginary circle. The pole should be placed clockwise from

1296-438: The circumscribed circle is: R = a 3 , {\displaystyle R={\frac {a}{\sqrt {3}}},} and the radius of the inscribed circle is half of the circumradius: r = 3 6 a . {\displaystyle r={\frac {\sqrt {3}}{6}}a.} The theorem of Euler states that the distance t {\displaystyle t} between circumradius and inradius

1350-437: The deltahedron and antiprism . It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry . An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on

1404-403: The precession of the equinoxes , the poles trace out circles on the celestial sphere, with a period of about 25,700 years. The Earth's axis is also subject to other complex motions which cause the celestial poles to shift slightly over cycles of varying lengths (see nutation , polar motion and axial tilt ). Finally, over very long periods the positions of the stars themselves change, because of

1458-419: The triangle inequality that the sum of any two of them is greater than the third. If P {\displaystyle P} is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem . A packing problem asks the objective of n {\displaystyle n} circles packing into

1512-407: The Earth (in contrast to Earth-centred, Earth-fixed frames), remaining always directed toward the equinox , and drifting over time with the motions of precession and nutation . In astronomy, there is also a heliocentric rectangular variant of equatorial coordinates, designated x , y , z , which has: This frame is similar to the ξ , η , ζ frame above, except that the origin

1566-442: The SMC and anticlockwise from the LMC. Going in the wrong direction will land you in the constellation of Horologium instead. A line from Sirius , the brightest star in the sky, through Canopus, the second-brightest, continued for the same distance lands within a couple of degrees of the pole. In other words, Canopus is halfway between Sirius and the pole. Equatorial coordinate system The equatorial coordinate system

1620-435: The area of a triangle is half the product of its base and height. The formula of the area of an equilateral triangle can be obtained by substituting the altitude formula. Another way to prove the area of an equilateral triangle is by using the trigonometric function . The area of a triangle is formulated as the half product of base and height and the sine of an angle. Because all of the angles of an equilateral triangle are 60°,

1674-425: The celestial sphere. The declination symbol δ , (lower case "delta", abbreviated DEC) measures the angular distance of an object perpendicular to the celestial equator, positive to the north, negative to the south. For example, the north celestial pole has a declination of +90°. The origin for declination is the celestial equator, which is the projection of the Earth's equator onto the celestial sphere. Declination

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1728-583: The centers of those equilateral triangles themselves form an equilateral triangle. Notably, the equilateral triangle tiles the Euclidean plane with six triangles meeting at a vertex; the dual of this tessellation is the hexagonal tiling . Truncated hexagonal tiling , rhombitrihexagonal tiling , trihexagonal tiling , snub square tiling , and snub hexagonal tiling are all semi-regular tessellations constructed with equilateral triangles. Other two-dimensional objects built from equilateral triangles include

1782-411: The constellation Musca is fairly easily recognised immediately beneath Crux. The second method uses Canopus (the second-brightest star in the sky) and Achernar . Make a large equilateral triangle using these stars for two of the corners. But where should the third corner go? It could be on either side of the line connecting Achernar and Canopus, and the wrong side will not lead to the pole. To find

1836-410: The correct side, imagine that Archernar and Canopus are both points on the circumference of a circle. The third corner of the equilateral triangle will also be on this circle. The corner should be placed clockwise from Achernar and anticlockwise from Canopus. The third imaginary corner will be the south celestial pole. If the opposite is done, the point will land in the middle of Eridanus , which isn't at

1890-587: The cross-section of the Gateway Arch and the surface of the Vegreville egg . It appears in the flag of Nicaragua and the flag of the Philippines . It is a shape of a variety of road signs , including the yield sign . The equilateral triangle occurs in the study of stereochemistry . It can be described as the molecular geometry in which one atom in the center connects three other atoms in

1944-472: The distance of the long axis in the direction the narrow end of the cross points, or join the two pointer stars with a line, divide this line in half, then at right angles draw another imaginary line through the sky until it meets the line from the Southern Cross. This point is 5 or 6 degrees from the south celestial pole. Very few bright stars of importance lie between Crux and the pole itself, although

1998-416: The equilateral triangles are regular polygons . The cevians of an equilateral triangle are all equal in length, resulting in the median and angle bisector being equal in length, considering those lines as their altitude depending on the base's choice. When the equilateral triangle is flipped across its altitude or rotated around its center for one-third of a full turn, its appearance is unchanged; it has

2052-440: The feet of the altitudes ), and the only triangle whose Steiner inellipse is a circle (specifically, the incircle). The triangle of the largest area of all those inscribed in a given circle is equilateral, and the triangle of the smallest area of all those circumscribed around a given circle is also equilateral. It is the only regular polygon aside from the square that can be inscribed inside any other regular polygon. Given

2106-438: The form 2 2 k + 1 , {\displaystyle 2^{2^{k}}+1,} wherein k {\displaystyle k} denotes the non-negative integer , and there are five known Fermat primes: 3, 5, 17, 257, 65537. A regular polygon is constructible by compass and straightedge if and only if the odd prime factors of its number of sides are distinct Fermat primes. To do so geometrically, draw

2160-491: The formula is as desired. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, for perimeter p {\displaystyle p} and area T {\displaystyle T} , the equality holds for the equilateral triangle: p 2 = 12 3 T . {\displaystyle p^{2}=12{\sqrt {3}}T.} The radius of

2214-499: The formula of an isosceles triangle by Pythagoras theorem : the altitude h {\displaystyle h} of a triangle is the square root of the difference of squares of a side and half of a base . Since the base and the legs are equal, the height is: h = a 2 − a 2 4 = 3 2 a . {\displaystyle h={\sqrt {a^{2}-{\frac {a^{2}}{4}}}}={\frac {\sqrt {3}}{2}}a.} In general,

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2268-401: The fundamental plane. This description of the orientation of the reference frame is somewhat simplified; the orientation is not quite fixed. A slow motion of Earth's axis, precession , causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic , completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the ecliptic, and

2322-450: The modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. The follow-up definition above may result in more precise properties. For example, since the perimeter of an isosceles triangle is the sum of its two legs and base, the equilateral triangle is formulated as three times its side. The internal angle of an equilateral triangle are equal, 60°. Because of these properties,

2376-469: The observer's meridian is said to have a zero hour angle (0 ). One sidereal hour (approximately 0.9973 solar hours ) later, Earth's rotation will carry the star to the west of the meridian, and its hour angle will be 1 . When calculating topocentric phenomena, right ascension may be converted into hour angle as an intermediate step. There are a number of rectangular variants of equatorial coordinates. All have: The reference frames do not rotate with

2430-611: The pole will have moved near the position of the star Alderamin (Alpha Cephei), and in 12,000 years, Vega (Alpha Lyrae) will become the "North Star", though it will be about six degrees from the true north celestial pole. To find Polaris, from a point in the Northern Hemisphere, face north and locate the Big Dipper (Plough) and Little Dipper asterisms. Looking at the "cup" part of the Big Dipper, imagine that

2484-472: The pole, but with a magnitude of 5.5 it is barely visible on a clear night. The south celestial pole can be located from the Southern Cross (Crux) and its two "pointer" stars α Centauri and β Centauri . Draw an imaginary line from γ Crucis to α Crucis —the two stars at the extreme ends of the long axis of the cross—and follow this line through the sky. Either go four-and-a-half times

2538-470: The pole. If Canopus has not yet risen, the second-magnitude Alpha Pavonis can also be used to form the triangle with Achernar and the pole. In this case, go anticlockwise from Achernar instead of clockwise, form the triangle with Canopus, and the third point, the pole, will reveal itself. The wrong way will lead to Aquarius, which is very far away from the celestial pole. The third method is best for moonless and clear nights, as it uses two faint "clouds" in

2592-506: The sides ( A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} being the vertices). There are numerous other triangle inequalities that hold equality if and only if the triangle is equilateral. The equilateral triangle can be constructed in different ways by using circles. The first proposition in the Elements first book by Euclid . Start by drawing

2646-432: The sides and altitude h {\displaystyle h} , d + e + f = h , {\displaystyle d+e+f=h,} independent of the location of P {\displaystyle P} . An equilateral triangle may have integer sides with three rational angles as measured in degrees, known for the only acute triangle that is similar to its orthic triangle (with vertices at

2700-744: The smallest possible equilateral triangle . The optimal solutions show n < 13 {\displaystyle n<13} that can be packed into the equilateral triangle, but the open conjectures expand to n < 28 {\displaystyle n<28} . Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. Viviani's theorem states that, for any interior point P {\displaystyle P} in an equilateral triangle with distances d {\displaystyle d} , e {\displaystyle e} , and f {\displaystyle f} from

2754-532: The stars also change slightly because of parallax effects). The north celestial pole currently is within one degree of the bright star Polaris (named from the Latin stella polaris , meaning " pole star "). This makes Polaris, colloquially known as the "North Star", useful for navigation in the Northern Hemisphere : not only is it always above the north point of the horizon, but its altitude angle

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2808-478: The stars' proper motions . To take into account such movement, celestial pole definitions come with an epoch to specify the date of the rotation axis; J2000.0 is the current standard. An analogous concept applies to other planets: a planet's celestial poles are the points in the sky where the projection of the planet's axis of rotation intersects the celestial sphere. These points vary because different planets' axes are oriented differently (the apparent positions of

2862-425: The symmetry of a dihedral group D 3 {\displaystyle \mathrm {D} _{3}} of order six. Other properties are discussed below. The area of an equilateral triangle with edge length a {\displaystyle a} is T = 3 4 a 2 . {\displaystyle T={\frac {\sqrt {3}}{4}}a^{2}.} The formula may be derived from

2916-669: The two stars at the outside edge of the cup form a line pointing upward out of the cup. This line points directly at the star at the tip of the Little Dipper's handle. That star is Polaris, the North Star. The south celestial pole is visible only from the Southern Hemisphere . It lies in the dim constellation Octans , the Octant. Sigma Octantis is identified as the south pole star, more than one degree away from

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