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Secant line

In geometry, a secant of a curve is a line that intersects the curve in at least two points. The word secant comes from the Latin word secare, meaning to cut. In the case of a circle, a secant will intersect the circle in two points and a chord is the line segment determined by these two points, the interval on a secant whose endpoints are these points. A straight line can intersect a circle in one or zero points. A line intersecting in two points is called a secant line, in one point a tangent line and in no points an exterior line. A chord of a circle is the line segment. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed by Euclid in his treatment, are proved. For example, Theorem: If C is a circle and ℓ a line that contains a point A, inside C and a point B, outside of C ℓ is a secant line for C. In some situations phrasing results in terms of secant lines instead of chords can help to unify statements.

As an example of this consider the result: If two secant lines contain chords AB and CD in a circle and intersect at a point P, not on the circle the line segment lengths satisfy AP⋅PB = CP⋅PD. If the point P lies inside the circle this is Euclid III.35, but if the point is outside the circle the result is not contained in the Elements. However, Robert Simson following Christopher Clavius demonstrated this result, sometimes called the secant-secant theorem, in their commentaries on Euclid. Working with curves more complicated than simple circles, the possibility that a line that meets the curve in two distinct points may meet the curve in further points arises; some authors define a secant line to a curve as a line. This definition leaves open the possibility that the line may have other points of intersection with the curve; when phrased this way the definitions of a secant line for circles and curves are identical and the possibility of additional points of intersection just does not occur for a circle.

Secants may be used to approximate the tangent line at some point P, if it exists. Define a secant to a curve by two points, P and Q, with P fixed and Q variable; as Q approaches P along the curve, if the slope of the secant approaches a limit value that limit defines the slope of the tangent line at P. The secant lines PQ are the approximations to the tangent line. In calculus, this idea is the geometric definition of the derivative. A tangent line to a curve at a point P may be a secant line to that curve if it intersects the curve in at least one point other than P. Another way to look at this is to realize that being a tangent line at a point P is a local property, depending only on the curve in the immediate neighborhood of P, while being a secant line is a global property since the entire domain of the function producing the curve needs to be examined; the concept of a secant line can be applied in a more general setting than Euclidean space. Let K be a finite set of k points in some geometric setting.

A line will be called an n-secant of K if it contains n points of K. For example, if K is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant and a line passing through only one of them would be a 1-secant. A unisecant in this example need not be a tangent line to the circle; this terminology is used in incidence geometry and discrete geometry. For instance, the Sylvester–Gallai theorem of incidence geometry states that if n points of Euclidean geometry are not collinear there must exist a 2-secant of them, and the original orchard-planting problem of discrete geometry asks for a bound on the number of 3-secants of a finite set of points. Finiteness of the set of points is not essential in this definition, as long as each line can intersect the set in only a finite number of points. Elliptic curve, a curve for which every secant has a third point of intersection, from which most of a group law may be defined Quadrisecant, a line that intersects four points of a curve Mean value theorem, that every secant of the graph of a smooth function has a parallel tangent line Secant plane, the three-dimensional equivalent of a secant line Secant variety, the union of secant lines and tangent lines to a given projective variety Weisstein, Eric W. "Secant line".

MathWorld

North Dakota Lottery

The North Dakota Lottery is run by the government of North Dakota. The Lottery began in 2004, following voter approval of an initiative constitutional amendment in 2002, Measure 2, which amended Article XI, Section 25 of the North Dakota Constitution to allow for the state to join a multi-state lottery "for the benefit of the State of North Dakota." In-state games were not allowed. As the Lottery is allowed only to offer multi-jurisdictional games, North Dakota-only games cannot be offered. All North Dakota Lottery games are part of the Multi-State Lottery Association. North Dakota's games are: Lucky for Life, Mega Millions, Lotto America, 2by2. North Dakota's lottery games require players to be at least 18 years old. 2by2 is offered in North Dakota and Nebraska. It is drawn nightly. 2by2 draws two red numbers from 1 through 26, two white numbers 1 through 26. Games cost $1 each. There are eight ways; the top prize is $22,000. In 2009, the Connecticut Lottery began a game known as Lucky-4-Life, it became a regional game.

After several modifications, in 2015 it became a "quasi-national" game. Lucky for Life plays are $2 each. Two number fields are used: players choose 5 of 48 "white balls", a green "Lucky Ball" from 18 numbers. Drawings remain in Connecticut. Top prize is $1,000-per-week-for-life. Winners of a "lifetime prize" can choose cash instead. Lucky for Life replaced Wild Card 2 in North Dakota in 2016. South Dakota and Idaho offered Wild Card 2. Lotto America is offered in 13 states. Lotto America draws five "red ball" numbers from 1 through 52, a "blue ball" numbered 1-10; the "All-Star Bonus" option multiplies non-jackpot prizes. Plays are $2 with the multiplier. Lotto America replaced Hot Lotto after the latter's final drawing on October 28, 2017. Powerball was North Dakota's first lottery game, its opening jackpot is $40 million. On October 13, 2009, the Mega Millions consortium and MUSL reached an agreement in principle to cross-sell Mega Millions and Powerball in U. S. lottery jurisdictions. The Lottery Advisory Commission voted unanimously on October 22, 2009 to bring Mega Millions to North Dakota, with Attorney General Wayne Stenehjem approving the game.

The Lottery joined Mega Millions on January 31, 2010. The current format for Mega Millions began on October 28, 2017. North Dakota Lottery website

Husam Badran

Husam Badran is the former leader of Hamas’s military wing in the northern West Bank. He was the orchestrator of several suicide bombings during the Second Intifada with the highest number of fatalities including the 2001 bombing which resulted in the Dolphinarium discotheque massacre in Tel Aviv which killed 21 people. Badran serves as the international spokesperson for Hamas using Twitter and news media to encourage Hamas militants to commit acts of political violence against Israelis and the Israeli government, he lives in Qatar. Badran was born on January 11, 1966 and is from Nablus, in the West Bank; the Second Intifada, or Palestinian uprising against Israel, was marked by widespread violence which erupted on Friday, September 28, 2000 and lasted until 2005. Protests began when Ariel Sharon visited Temple Mount in the Old City Jerusalem with 1,000 Israeli police in what Palestinians saw as a blatant attempt to provoke them. Protests turned into terrorism targeting Israeli civilians on buses, on city streets.

During this period, Husam Badran was the commander of Hamas’s military wing in the Samaria area and was involved in orchestrating the suicide bombing on Sbarro Pizza, the Dolphinarium Discotheque bombing, the bombing of Passover seder at the Park Hotel, the bombing of Matza restaurant in Haifa as part of the Second Intifada attacks. More than 100 people were killed in Second Intifada terrorist attacks ordered by Badran. Badran was arrested in 2002 in as a part of Operation “Defensive Shield," a large scale military operation initiated by the Israeli Defense Forces against Palestinian Militants in the West Bank. In 2004, he was sentenced to 17 years of imprisonment for his role in the Second Intifada, but he was expelled to Qatar. Badran took part in orchestrating the 2001 bombing of Sbarro Pizza in Jerusalem, which led to the death of 15 Israelis and injured 130 others; the attack, which took place on August 9 at 2:00pm, was carried out by a lone suicide bomber who transported an explosive device, enhanced with screws and nails, in a guitar case and entered the crowded restaurant filled with mothers and children.

Just before midnight on Friday June 1, a suicide bomber stood in line with a large group of teenagers waiting to get inside a disco. While in line, the bomber detonated the explosives attached to his body killing 21 people and wounding 120 others; the explosive contained nails in an effort to increase the extent of injuries. Most of the murdered teens were from the former Soviet Union and had planned to attend a dance party inside the Dolphinarium Disco. Other young people standing in line at the adjacent nightclub, were killed in the attack. 17 people were killed in the suicide bomb attack, another 4 died after succumbing to their injuries. On March 27, 2002 a suicide bomber entered the Park Hotel in the coastal city of Netanya where 250 guests were celebrating the Passover Holiday; the terrorist walked into the dining room of the hotel and detonated his explosive. Thirty people were killed in the explosion and 140 - 20 of them, seriously. On March 31, 2002, in the crowded Matza restaurant in the northern Israeli city of Haifa, a suicide bomber set off an explosion, killing himself and at least 14 other people.

At least 33 people were taken to hospitals. Three of the injured were in critical condition. At least 14 people were killed at another person died during surgery. Since his deportation from Palestine, Badran has been based in Qatar where he acts as a media spokesman for the Hamas movement emailing statements to various news agencies to register official Hamas positions on Middle Eastern news, offering support for violent actions orchestrated by Hamas and other Palestinian militants, expressing dissent against Israeli military activities, he makes use of social media to express support of Hamas and to decry the Israeli government. While his Facebook page has been removed for reasons unstated, his Twitter page remains active. According to the IDF, Badran has been active in sending money transfers totaling hundreds of thousands of dollars to Palestine in an effort to fund the Hamas terror network in and around Nablus in the Northern West Bank. IDF operations by 2016 had thwarted terror attacks funded by Badran.

He is responsible for announcing Qatar’s uninterrupted financial support of Hamas