Sine-triple-angle circle

Last updated September 19, 2024

In triangle geometry, the sine-triple-angle circle is one of a circle of the triangle.^[1]^[2] Let $A 1$ and $A 2$ points on $BC$ , a side of triangle $ABC$ . And, define $B 1, B 2, C 1$ and $C 2$ similarly for $CA$ and $AB$ . If

Properties

$|A_{1}A_{2}|:|B_{1}B_{2}|:|C_{1}C_{2}|=\sin 3A:\sin 3B:\sin 3C$ .^[5] This property is the reason why the circle called "sine-triple-angle circle". But, the number of circle which cuts three sides of triangle that satisfies the ratio are countless. The centers of these circles are on the hyperbola through the incenter, three excenters, and X(49) (see below for X₄₉).^[6]
The homothetic centers of Nine-point circle and the circle are the Kosnita point and the focus of Kiepert parabola.
The homothetic centers of circumcircle and the circle are X(184), the inverse of Jerabek center in Brocard circle, and X(1147).^[7]
Intersections of Polar of $A,B$ and $C$ with the circle and $BC,CA$ and $AB$ are colinear.^[8]
The radius of sine-triple-angle circle is

${\frac {R}{|1+8\cos(A)\cos(B)\cos(C)|}},$

where $R$ is the circumradius of triangle $ABC$ .

Center

The center of sine-triple-angle circle is a triangle center designated as X(49) in Encyclopedia of Triangle Centers.^[7]^[9] The trilinear coordinates of X(49) is

$\cos(3A):\cos(3B):\cos(3C)$ .

Generalization

For natural number n>0, if

$\angle A_{1}C_{1}A_{2}=(2n-1)A-(n-1)\pi ,$

$\angle B_{1}A_{1}B_{2}=(2n-1)B-(n-1)\pi ,$

and

$\angle C_{1}B_{1}C_{2}=(2n-1)C-(n-1)\pi ,$

then $A 1, A 2, B 1, B 2, C 1$ and $C 2$ are concyclic.^[8] Sine-triple-angle circle is the special case in n=2.

Also,

$|A_{1}A_{2}|:|B_{1}B_{2}|:|C_{1}C_{2}|=\sin(2n-1)A:\sin(2n-1)B:\sin(2n-1)C$ .

Related Research Articles

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points $(cos t, sin t)$ form a circle with a unit radius, the points $(cosh t, sinh t)$ form the right half of the unit hyperbola. Also, similarly to how the derivatives of $sin(t)$ and $cos(t)$ are $cos(t)$ and $-sin(t)$ respectively, the derivatives of $sinh(t)$ and $cosh(t)$ are $cosh(t)$ and $+sinh(t)$ respectively.

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, $where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles, while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals; The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data and the technique gives two possible values for the enclosed angle.$

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

<span class="mw-page-title-main">Pedal curve</span> Curve generated by the projections of a fixed point on the tangents of another curve

In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal pointP, the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X. Conversely, at any point R on the curve C, let T be the tangent line at that point R; then there is a unique point X on the tangent T which forms with the pedal point P a line perpendicular to the tangent T – the pedal curve is the set of such points X, called the foot of the perpendicular to the tangent T from the fixed point P, as the variable point R ranges over the curve C.

The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all the trisectors are intersected, one obtains four other equilateral triangles.

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

In geometry, the area enclosed by a circle of radius $r$ is $π r 2$ . Here the Greek letter $π$ represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius b with centers distance 2a apart. A line segment of length 2c attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as the circles rotate partially back and forth or completely around. It arose in connection with James Watt's pioneering work on the steam engine.

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $, the sine and cosine functions are denoted as and .$

$<span class="mw-page-title-main">Exact trigonometric values</span> Trigonometric values in terms of square roots and fractions$

In mathematics, the values of the trigonometric functions can be expressed approximately, as in $, or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.$

Trigonometry is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios such as sine.

In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides $and opposite respective angles and, the law of cosines states:$

In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point $P$ is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

<span class="mw-page-title-main">Pythagorean theorem</span> Relation between sides of a right triangle

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as $S 1$ because it is a one-dimensional unit $n$ -sphere.

<span class="mw-page-title-main">Integer triangle</span> Triangle with integer side lengths

An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles.

References

↑ Mathworld,Weisstein, Eric W
↑ Society, London Mathematical (1893). Proceedings of the London Mathematical Society. Oxford University Press. p. 162.
↑ The Messenger of Mathematics. Macmillan and Company. 1887. p. 125.
↑ Mathesis (in French). Vol. 7. Johnson Reprint Corporation. 1964.
↑ Thebault (1956)
↑ Ehrmann and van Lamoen (2002)
1 2 "Clark Kimberling's rightri Encyclopedia of Triangle Centers - ETC".
1 2 Mathematical Questions and Solutions. F. Hodgson. 1887. p. 139.
↑ Congressus Numerantium. Utilitas Mathematica Pub. Incorporated. 1970.

V. Thebault (1965). Sine-triple-angle-circle. Vol. 65. Mathesis. pp. 282–284.
Ehrmann, Jean-Pierre; Lamoen, Floor van (2002). The Stammler Circles. Forum Geometricorum. pp. 151–161.

External links

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Mathworld,Weisstein, Eric W

[2] Society, London Mathematical (1893). Proceedings of the London Mathematical Society. Oxford University Press. p. 162.

[3] The Messenger of Mathematics. Macmillan and Company. 1887. p. 125.

[4] Mathesis (in French). Vol. 7. Johnson Reprint Corporation. 1964.

[5] Thebault (1956)

[6] Ehrmann and van Lamoen (2002)

[:0-7] 1 2 "Clark Kimberling's rightri Encyclopedia of Triangle Centers - ETC".

[:1-8] 1 2 Mathematical Questions and Solutions. F. Hodgson. 1887. p. 139.

[9] Congressus Numerantium. Utilitas Mathematica Pub. Incorporated. 1970.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]