Lines that intersect Circles
... Chord: is a segment whose endpoints lie on a circle. Diameter: -a chord that contains the center -connects two points on the circle and passes through the center Secant: line that intersects a circle at two points ...
... Chord: is a segment whose endpoints lie on a circle. Diameter: -a chord that contains the center -connects two points on the circle and passes through the center Secant: line that intersects a circle at two points ...
Girard`s Theorem: Triangles and
... declare the π version of the formula to be much more elegant and ‘natural’. But we might instead say “How suggestive that τ/2 is! What result of circle or spherical geometry does that come from?” And this is a natural and elegant question to ask! The proof of Girard’s theorem is rather easy once we ...
... declare the π version of the formula to be much more elegant and ‘natural’. But we might instead say “How suggestive that τ/2 is! What result of circle or spherical geometry does that come from?” And this is a natural and elegant question to ask! The proof of Girard’s theorem is rather easy once we ...
Chapter 1 – Points, Lines, Planes, and Angles
... one side of AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: a) OA is paired with 0 and OB with 180. b) If OP is paired with x, and OQ with y, then m
... one side of AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: a) OA is paired with 0 and OB with 180. b) If OP is paired with x, and OQ with y, then m
Course Outlines
... *G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove ...
... *G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove ...
Chapter 1
... A ray, line, or segment that divides an angle into two congruent angles. Any line, ray, or segment that passes through the midpoint of the segment. A segment bisector divides a segment into two congruent segments. Collinear points are points that lie on the same line. Non-collinear points are points ...
... A ray, line, or segment that divides an angle into two congruent angles. Any line, ray, or segment that passes through the midpoint of the segment. A segment bisector divides a segment into two congruent segments. Collinear points are points that lie on the same line. Non-collinear points are points ...
chapter 3 topics
... Theorem 3.12 Lines Perpendicular to a Transversal Theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. ...
... Theorem 3.12 Lines Perpendicular to a Transversal Theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. ...
TWO CAMERAS 2009
... closeness to the human perceptions. Although the laws of the central projection and perspective are known for many years, the mathematicians investigate them now in the framework of the projective, affine and Euclidian geometry to obtain more efficient and elegant results. In this context we present ...
... closeness to the human perceptions. Although the laws of the central projection and perspective are known for many years, the mathematicians investigate them now in the framework of the projective, affine and Euclidian geometry to obtain more efficient and elegant results. In this context we present ...
Alliance Cindy and Bill Simon Technology Academy Digital Agenda
... 1. In your group of 2, use Geogebra to construct a circle and a diameter. 2. Construct a point on one of the semicircles, and construct two chords from it to the endpoints of the diameter to create a right triangle. 3. Locate the midpoint of each of the two chords. 4. Predict, then sketch the locus ...
... 1. In your group of 2, use Geogebra to construct a circle and a diameter. 2. Construct a point on one of the semicircles, and construct two chords from it to the endpoints of the diameter to create a right triangle. 3. Locate the midpoint of each of the two chords. 4. Predict, then sketch the locus ...
Chapter 1
... Points and lines that are in the same plane. A, B, C are coplanar. Points and lines that are NOT on the same plane are said to be ...
... Points and lines that are in the same plane. A, B, C are coplanar. Points and lines that are NOT on the same plane are said to be ...
Lie sphere geometry
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous ""line-sphere correspondence"" between the space of lines and the space of spheres in 3-dimensional space.