2-1 - Lee County School District

... Unit 4- Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of ...

... Unit 4- Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of ...

CHAPTER 1. LINES AND PLANES IN SPACE §1. Angles and

... 1.16. We may assume that the given lines pass through one point. Let a1 and a2 be the bisectors of the angles between the first and the second line, b1 and b2 the bisectors between the second and the third lines. A line forms equal angles with the three given lines if and only if it is perpendicular ...

... 1.16. We may assume that the given lines pass through one point. Let a1 and a2 be the bisectors of the angles between the first and the second line, b1 and b2 the bisectors between the second and the third lines. A line forms equal angles with the three given lines if and only if it is perpendicular ...

LSU College Readiness Program COURSE

... and pencil work. Requires paper and pencil work. Requires paper and pencil work. Requires paper and pencil work. ...

... and pencil work. Requires paper and pencil work. Requires paper and pencil work. Requires paper and pencil work. ...

Exploring Advanced Euclidean Geometry

... The geometry studied in this book is Euclidean geometry. Euclidean geometry is named for Euclid of Alexandria, who lived from approximately 325 BC until about 265 BC. The ancient Greeks developed geometry to a remarkably advanced level and Euclid did his work during the later stages of that developm ...

... The geometry studied in this book is Euclidean geometry. Euclidean geometry is named for Euclid of Alexandria, who lived from approximately 325 BC until about 265 BC. The ancient Greeks developed geometry to a remarkably advanced level and Euclid did his work during the later stages of that developm ...

Plane Geometry - UVa-Wise

... The opening lines in the subject of geometry were written around 300 B.C. by the Greek mathematician Euclid in 13 short books gathered into a collection called The Elements. Now certainly geometry existed before Euclid, often in a quite sophisticated form. It arose from such practical concerns as pa ...

... The opening lines in the subject of geometry were written around 300 B.C. by the Greek mathematician Euclid in 13 short books gathered into a collection called The Elements. Now certainly geometry existed before Euclid, often in a quite sophisticated form. It arose from such practical concerns as pa ...

developing auxiliary resource materials

... In recent years, teachers of mathematics had numerous resources upon which they could refer to when designing their daily, weekly, even yearly units and lesson plans. In most cases, a curriculum was determined and resources were purchased by the state or local districts to be distributed to the scho ...

... In recent years, teachers of mathematics had numerous resources upon which they could refer to when designing their daily, weekly, even yearly units and lesson plans. In most cases, a curriculum was determined and resources were purchased by the state or local districts to be distributed to the scho ...

non-euclidean geometry

... Figure 5.1 Triangle in standard position...........................................................… ...........86 Figure 5.2 Singly asymptotic triangle ABZ.........................................................… ........88 Figure 5.3 The triangle as the difference of two singly asymptotic triangl ...

... Figure 5.1 Triangle in standard position...........................................................… ...........86 Figure 5.2 Singly asymptotic triangle ABZ.........................................................… ........88 Figure 5.3 The triangle as the difference of two singly asymptotic triangl ...

IMO problems from Kalva `s Web

... is colored red or blue, so that if (h, k) is red and h' ≤ h, k' ≤ k, then (h', k') is also red. A type 1 subset of S has n blue elements with different first member and a type 2 subset of S has n blue elements with different second member. Show that there are the same number of type 1 and type 2 sub ...

... is colored red or blue, so that if (h, k) is red and h' ≤ h, k' ≤ k, then (h', k') is also red. A type 1 subset of S has n blue elements with different first member and a type 2 subset of S has n blue elements with different second member. Show that there are the same number of type 1 and type 2 sub ...

# Problem of Apollonius

In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (ca. 262 BC – ca. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, ""Tangencies""); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2) and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed (its complement is excluded) and there are 8 subsets of a set whose cardinality is 3, since 8 = 23.In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN.Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.Apollonius' problem has stimulated much further work. Generalizations to three dimensions—constructing a sphere tangent to four given spheres—and beyond have been studied. The configuration of three mutually tangent circles has received particular attention. René Descartes gave a formula relating the radii of the solution circles and the given circles, now known as Descartes' theorem. Solving Apollonius' problem iteratively in this case leads to the Apollonian gasket, which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy–Littlewood circle method.