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Rhombus

In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length; the rhombus is called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers to a rhombus with a 60° angle, the latter sometimes refers to a rhombus with a 45° angle. Every rhombus is simple, is a special case of a parallelogram and a kite. A rhombus with right angles is a square; the word "rhombus" comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning "to turn round and round." The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for a bicone, two right circular cones sharing a common base. The surface we refer to as rhombus today is a cross section of the bicone on a plane through the apexes of the two cones. A simple quadrilateral is a rhombus if and only if it is any one of the following: a parallelogram in which a diagonal bisects an interior angle a parallelogram in which at least two consecutive sides are equal in length a parallelogram in which the diagonals are perpendicular a quadrilateral with four sides of equal length a quadrilateral in which the diagonals are perpendicular and bisect each other a quadrilateral in which each diagonal bisects two opposite interior angles a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, DAP are all congruent a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point Every rhombus has two diagonals connecting pairs of opposite vertices, two pairs of parallel sides.

Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: Opposite angles of a rhombus have equal measure; the two diagonals of a rhombus are perpendicular. Its diagonals bisect opposite angles; the first property implies. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel, thus denoting the common side as a and the diagonals as p and q, in every rhombus 4 a 2 = p 2 + q 2. Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of, a line of symmetry, is a kite; every rhombus is a kite, any quadrilateral, both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral; that is, it has an inscribed circle, tangent to all four sides. The length of the diagonals p = AC and q = BD can be expressed in terms of the rhombus side a and one vertex angle α as p = a 2 + 2 cos ⁡ α and q = a 2 − 2 cos ⁡ α.

These formulas are a direct consequence of the law of cosines. The inradius, denoted by r, can be expressed in terms of the diagonals p and q as r = p ⋅ q 2 p 2 + q 2, or in terms of the side length a and any vertex angle α or β as r = a sin ⁡ α 2 = a sin ⁡ β 2; as for all parallelograms, the area K of a rhombus is the product of its height. The base is any side length a: K = a ⋅ h; the area can be expressed as the base squared times the sine of any angle: K = a 2 ⋅ sin ⁡ α = a 2 ⋅ sin ⁡ β, or in terms of the height and a vertex angle: K = h 2 sin ⁡ α, or as half the product of the diagonals p, q: K = p ⋅ q 2, or as the semiperimeter times the radius of the circle inscribed in the rhombus: K = 2 a ⋅ r. Another way, in common with parallelograms, is to consider t

Indian black-lored tit

The Indian black-lored tit, Indian tit, or Indian yellow tit is a passerine bird in the tit family Paridae. The yellow-cheeked tit is its closest relative, both may be related to the yellow tit; these three tits certainly form a distinct lineage as evidenced by morphology, mtDNA cytochrome b sequence analysis. The subgenus name Macholophus may apply for them; this species is a resident breeder on the Indian subcontinent. It does not occur in Sri Lanka, it is an active and agile feeder, taking insects and spiders from the canopy, sometimes fruit. It is an easy tit to recognise in most of India, large in size at 13 cm, with a broad black line down its otherwise yellow front; the large crest, neck and head are black with yellow cheeks and supercilia. Upperparts are olive-green, it has two white or yellowish wingbars and white outer tail feathers. Females and young birds are duller than males; the underpart colour becomes dull from north to south through this tit's range. It is, like other tits, a vocal bird, has a large variety of calls, of which the most familiar is a si-si.

The song is a sometimes nuthatch-like chi-chi-chi. Woodpecker or barbet holes are used for a nest, this species will excavate its own hole or use man-made sites; the clutch is 3-5 white eggs, spotted red. The bird is a close sitter; the Indian black-lored tit was one of the many species in the genus Parus but was moved to Machlolophus after a molecular phylogenetic analysis published in 2013 showed that the members of the new genus formed a distinct clade. Gill, Frank B.. "Phylogeny of titmice: II. Species relationships based on sequences of the mitochondrial cytochrome-b gene". Auk. 122: 121–143. Doi:10.1642/0004-80381222.0. CO. Grimmett, Richard. Princeton University Press, Princeton, N. J.. ISBN 0-691-04910-6 Harrap, Simon & Quinn, David: Tits, Nuthatches & Treecreepers. Christopher Helm, London. ISBN 0-7136-3964-4 Rasmussen, P. C. and J. C. Anderton. 2005. Birds of South Asia; the Ripley guide. Volume 2: attributes and status. Smithsonian Institution and Lynx Edicions, Washington D. C. and Barcelona. Grimmett, Richard.

Oxford University Press. ISBN 0-19-807722-X

Thomas R. Fitzgerald (judge)

Thomas R. Fitzgerald was a chief justice of the Illinois Supreme Court, he became the first Illinois chief justice to preside over the impeachment trial of a sitting governor when he presided over the impeachment trial of Governor Rod Blagojevich. Born in Chicago on July 10, 1941, Fitzgerald graduated from Leo Catholic High School in 1959, he attended Loyola University Chicago before enlisting in the United States Navy. Following his tour of duty in the Navy, he graduated with honors from The John Marshall Law School, where he was a founder of the school's current law review and served as its associate editor; the son of a circuit court judge, Fitzgerald began his own career in the law as a prosecutor in the Cook County State's Attorney's Office. When first elected to the bench in 1976, he was the youngest Cook County judge, he served as a trial judge in the Criminal Court from 1976 to 1987, when he was assigned Supervising Judge of Traffic Court. In 1989, he returned to the Criminal Division as Presiding judge.

He was appointed to serve as presiding judge of Illinois's first statewide Grand Jury. He was elected to the Supreme Court of Illinois for the First District in 2000; as a law professor, Fitzgerald taught at The John Marshall Law School and Chicago-Kent College of Law, where he was assistant coordinator of the trial advocacy program from 1986 to 1996. He has taught at the Einstein Institute for Science and the Courts. Fitzgerald served as president of the Illinois Judges' Association, chair of the Illinois Supreme Court Special Committee on Capital Cases, member of the Governor's Task Force on Crime and Corrections, chairman of several committees of the Illinois Judicial Conference, member of the Chicago Bar Association's Board of Managers and past chairman of the Chicago Bar Association's committees on constitutional law and long-range planning, he was elected Chief Justice of the Illinois Supreme Court on May 19, 2008, with his term effective September 6, 2008. On January 26, 2009, he became the first Illinois chief justice to preside over the impeachment trial of a sitting governor, as he opened proceedings in the impeachment trial of Governor Rod Blagojevich.

Fitzgerald retired in October, 2010. He announced his retirement on September 2010, citing a recent diagnosis of Parkinson's Disease. Fitzgerald died on November 1, 2015 at his home from the disease at the age of 74. Fitzgerald has been awarded the Outstanding Judicial Performance Award by the Chicago Crime Commission and honored as Celtic Man of the Year by the Celtic Legal Society, he received the Herman Kogan Media Award for Excellence in Broadcast Journalism. The Lawyers' Assistance Program honored him in 2000 with the Hon. John Powers Crowley Award, he is the 2001 recipient of the John Marshall Law School Freedom Award. In 2003, Fitzgerald was awarded the Joel Flaum Award by the Chicago Inn of Court and the Chicago-Kent College of Law Professional Achievement Award. In 2005, Fitzgerald was named Catholic Lawyer of the Year by the Catholic Lawyers Guild of Chicago. In 2008, he was awarded the John Paul Stevens Award by the Chicago Bar Association and the Chicago Bar Foundation, he is a member of the Leo High School Hall of Fame.

Fitzgerald was named Chicago Lawyer's 2010 Person of the Year