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Semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter . Motivation: triangles The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths :s = \frac. Properties In any triangle, any vertex and the point where the opposite excircle touches the triangle partition the triangle's perimeter into two equal lengths, thus creating two paths each of which has a length equal to the semiperimeter. If are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (, shown in red in the diagram) are known as splitters, and :\begin s &= , AB, +, A'B, =, AB, +, AB', =, AC, +, A'C, \\ &= , AC, +, AC', =, BC, +, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heron's Formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work ''Metrica'', though it was probably known centuries earlier. Example Let be the triangle with sides , , and . This triangle's semiperimeter is s = \tfrac12(a+b+c)= \tfrac12(4+13+15) = 16 therefore , , , and the area is \begin A &= \\ mu&= \\ mu&= 24. \end In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number. Alternate expressions Heron's formula can also be written in terms of just ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extended side, extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal and external angle, internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the internal and external angle, external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the Incircle and excircles of a triangle, incircle of the quadrilateral or its inscribed circle, its center is the ''incenter'' and its radius is called the ''inradius''. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called ''circumscribable quadrilaterals'', ''circumscribing quadrilaterals'', and ''circumscriptible quadrilaterals''. Tangential quadrilaterals are a special case of tangential polygons. Other less frequently used names for this class of quadrilaterals are ''inscriptable quadrilateral'', ''inscriptible quadrilateral'', ''inscribable quadrilateral'', ''circumcyclic quadrilateral'', and ''co-cyclic quadrilateral''.. Due to the risk of confusion with a qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Quadrilateral
In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''concyclic''. The center of the circle and its radius are called the ''circumcenter'' and the ''circumradius'' respectively. Usually the quadrilateral is assumed to be convex polygon, convex, but there are also Crossed quadrilateral, crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case. The word cyclic is from the Ancient Greek (''kuklos''), which means "circle" or "wheel". All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section Cyclic quadrilateral#Characterizations, characterizations below states what necessar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta's Formula
In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, ''Bretschneider's formula'', can be used with non-cyclic quadrilateral. ''Heron's formula'' can be thought as a special case of the Brahmagupta's formula for triangles. Formulation Brahmagupta's formula gives the area of a convex cyclic quadrilateral whose sides have lengths , , , as : K=\sqrt where , the semiperimeter, is defined to be : s=\frac. This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as (or any one side) approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula. If the semiperimeter is not used, Bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either simple polygon, simple (not self-intersecting), or complex polygon, complex (self-intersecting, or crossed). Simple quadrilaterals are either convex polygon, convex or concave polygon, concave. The Internal and external angle, interior angles of a simple (and Plane (geometry), planar) quadrilateral ''ABCD'' add up to 360 Degree (angle), degrees, that is :\angle A+\angle B+\angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicentric Quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a ''double circle quadrilateral'' and ''double scribed quadrilateral''. If two circles, one within the other, are the incircle and the circumcircle of a bicentric quadrilateral, then every point on the circumcircle is the vertex of a bicentric quadrilateral having the same incircle and circumcircle. This is a special case of Poncelet's porism, which was proved by the French mathematician Jean-Victor Poncelet (1788–1867). Special cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cotangents
In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. Just as three quantities whose equality is expressed by the law of sines are equal to the diameter of the circumscribed circle of the triangle (or to its reciprocal, depending on how the law is expressed), so also the law of cotangents relates the radius of the inscribed circle of a triangle (the inradius) to its sides and angles. Statement Using the usual notations for a triangle (see the figure at the upper right), where are the lengths of the three sides, are the vertices opposite those three respective sides, are the corresponding angles at those vertices, is the semiperimeter, that is, , and is the radius of the inscribed circle, the law of trigonometric function, cotangents states that \frac = \frac = \frac = \frac, and furthermore that the inradius is given by r = \sqrt\,. Proof In the upper figure, the points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bretschneider's Formula
In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals, whether it is cyclic or not. The formula also works on crossed quadrilaterals provided that directed angles are used. History The German mathematician Carl Anton Bretschneider discovered the formula in 1842. The formula was also derived in the same year by the German mathematician Karl Georg Christian von Staudt. Formulation Bretschneider's formula is expressed as: : K = \sqrt ::= \sqrt . Here, , , , are the sides of the quadrilateral, is the semiperimeter, and and are any two opposite angles, since \cos (\alpha+ \gamma) = \cos (\beta+ \delta) as long as directed angles are used so that \alpha+\beta+\gamma+\delta=360^ or \alpha+\beta+\gamma+\delta=720^ (when the quadrilateral is crossed). Proof Denote the area of the quadrilateral by . Then we have : \begin K &= \frac + \frac.\end Therefore : 2K= (ad) \sin \a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle is called the '' hypotenuse'' (side c in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: '' cathetus''). Side a may be identified as the side ''adjacent'' to angle B and ''opposite'' (or ''opposed to'') angle A, while side b is the side adjacent to angle A and opposite angle B. Every right triangle is half of a rectangle which has been divided along its diagonal. When the rectangle is a square, its right-triangular half is isosceles, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is scalene. Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
