Circles defined for a triangle

Polar circle (geometry)

In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively. The first parts of the radius formula reflect the fact that the orthocenter divides the altitudes into segment pairs of equal products. The trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse, so one of its angles is obtuse and hence has a negative cosine. (Wikipedia).

Polar circle (geometry)
Video thumbnail

Calculus 2: Polar Coordinates (1 of 38) What are Polar Coordinates?

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what are polar coordinates and Cartesian coordinates. The Cartesian coordinates use x and y to locate a point on a plane, and the polar coordinates use r and theta to locate a point on a plane

From playlist THE "WHAT IS" PLAYLIST

Video thumbnail

Polar Coordinates and Graphing Polar Equations

Everything we have done on the coordinate plane so far has been using rectangular coordinates. That's the x and y we are used to. But that's not the only coordinate system. We can also use polar coordinates, which graph points in terms of a radius, or distance from a pole, and theta, the a

From playlist Mathematics (All Of It)

Video thumbnail

Calculus 10.3 Polar Coordinates

My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart

From playlist Calculus

Video thumbnail

Calculus 2 Lecture 10.5: Calculus of Polar Equations

Calculus 2 Lecture 10.5: Calculus of Polar Equations. Area Bound by Polar Curve, Area Between Two Polar Curves.

From playlist Calculus 2 (Full Length Videos)

Video thumbnail

Calculus 2: Polar Coordinates (19 of 38) Area Bounded by a Polar Curve

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the theory and develop the equation to find the area A=? bounded by a polar curve. Next video in the series can be seen at: https://youtu.be/-7Tp8WM2d2Q

From playlist CALCULUS 2 CH 10 POLAR COORDINATES

Video thumbnail

Ex: Find the Polar Equation of a Circle With Center at the Origin

This video explains how to determine the equation of a circle in rectangular form and polar form from the graph of a circle centered at the origin. Library: http://www.mathispower4u.com Search: http://www.mathispower4u.wordpress.com

From playlist Polar Coordinates and Equations

Video thumbnail

Quickly fill in the unit circle by understanding reference angles and quadrants

👉 Learn about the unit circle. A unit circle is a circle which radius is 1 and is centered at the origin in the cartesian coordinate system. To construct the unit circle we take note of the points where the unit circle intersects the x- and the y- axis. The points of intersection are (1, 0

From playlist Trigonometric Functions and The Unit Circle

Video thumbnail

Introduction to Polar Coordinates

This video introduces polar coordinates http://mathispower4u.wordpress.com/

From playlist Polar Coordinates and Equations

Video thumbnail

Apollonius and polarity | Universal Hyperbolic Geometry 1 | NJ Wildberger

This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical projective geometry and the study of a distinguished circle. This subject will be called Universal Hyperbolic Geometry, as it extends t

From playlist Universal Hyperbolic Geometry

Video thumbnail

Duality: magic in simple geometry #SoME2

Two inaccuracies: 2:33 explains the first property (2:16), not the second one (2:24) Narration at 5:52 should be "intersections of GREEN and orange lines" Time stamps: 0:00 — Intro 0:47 — Polar transform 4:46 — Desargues's Theorem 6:29 — Pappus's Theorem 7:18 — Sylvester-Gallai Theorem 8

From playlist Summer of Math Exposition 2 videos

Video thumbnail

Mumford-Tate Groups and Domains - Phillip Griffiths

Phillip Griffiths Professor Emeritus, School of Mathematics March 28, 2011 For more videos, visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

A brief history of Geometry III: The 19th century | Sociology and Pure Mathematics | N J Wildberger

The 19th century was a pivotal time in the development of modern geometry, actually a golden age for the subject, which then saw a precipitous decline in the 20th century. Why was that? To find out, let's first overview some of the main developments in geometry during the 1800's, includin

From playlist Sociology and Pure Mathematics

Video thumbnail

Perpendicularity, polarity and duality on a sphere | Universal Hyperbolic Geometry 37

This video discusses perpendicularity on a sphere, associating two poles to every great circle, and one polar line (great circle) to every point. This association is cleaner in elliptic geometry, where there is then a 1-1 correspondence between elliptic points (pairs of antipodal points on

From playlist Universal Hyperbolic Geometry

Video thumbnail

Mathematics of Animation

In the past few years, animations on the web have evolved dramatically. We are able to build animations that generate and manipulate complex geometry, react to user input and tell captivating stories. Varun finds himself looking at all this amazing work and asking, “How did they do that?”

From playlist Web Design: CSS / SVG

Video thumbnail

MIT 3.60 | Lec 7b: Symmetry, Structure, Tensor Properties of Materials

Part 2: 2D Plane Groups, Lattices (cont.) View the complete course at: http://ocw.mit.edu/3-60F05 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 3.60 Symmetry, Structure & Tensor Properties of Material

Video thumbnail

The Three/Four bridge and Apollonius duality for conics | Six: A course in pure maths 5 | Wild Egg

The Three / Four bridge plays an important role in understanding the remarkable duality discover by Apollonius between points and lines in the plane once a conic is specified. This is a purely projective construction that works for ellipses, and their special case of a circle, for parabola

From playlist Six: An elementary course in Pure Mathematics

Video thumbnail

S.A.Robertson, How to see objects in four dimensions, LMS 1993

Based on the 1993 London Mathematical Society Popular Lectures, this special 'television lecture' is entitled "How to see objects in four dimensions" by Professor S.A.Robertson. The London Mathematical Society is one of the oldest mathematical societies, founded in 1865. Despite it's name

From playlist Mathematics

Video thumbnail

What is the unit circle

👉 Learn about the unit circle. A unit circle is a circle which radius is 1 and is centered at the origin in the cartesian coordinate system. To construct the unit circle we take note of the points where the unit circle intersects the x- and the y- axis. The points of intersection are (1, 0

From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)

Video thumbnail

Animation: Comparing Polar and Rectangular Coordinates

This animation compares points plotted using polar and rectangular coordinates. http://mathispower4u.wordpress.com/

From playlist Polar Equations

Video thumbnail

Cosmology Lecture 3

(January 28, 2013) Leonard Susskind presents three possible geometries of homogeneous space: flat, spherical, and hyperbolic, and develops the metric for these spatial geometries in spherical coordinates. Originally presented in the Stanford Continuing Studies Program. Stanford Universit

From playlist Lecture Collection | Cosmology

Related pages

Cosine | Vertex (geometry) | Tangential triangle | Orthocenter | Geometry | Angle | Triangle | Nine-point circle | Circle | Circumscribed circle | Orthocentric system | Trigonometry