In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C respectively. These three reflected lines concur at the isogonal conjugate of P. (This definition applies only to points not on a sideline of triangle ABC.) This is a direct result of the trigonometric form of Ceva's theorem. The isogonal conjugate of a point P is sometimes denoted by P*. The isogonal conjugate of P* is P. The isogonal conjugate of the incentre I is itself. The isogonal conjugate of the orthocentre H is the circumcentre O. The isogonal conjugate of the centroid G is (by definition) the symmedian point K. The isogonal conjugates of the Fermat points are the isodynamic points and vice versa. The Brocard points are isogonal conjugates of each other. In trilinear coordinates, if X = x : y : z is a point not on a sideline of triangle ABC, then its isogonal conjugate is 1⁄x : 1⁄y : 1⁄z. For this reason, the isogonal conjugate of X is sometimes denoted by X–1. The set S of triangle centers under trilinear product, defined by , is a commutative group, and the inverse of each X in S is X–1. As isogonal conjugation is a function, it makes sense to speak of the isogonal conjugate of sets of points, such as lines and circles. For example, the isogonal conjugate of a line is a circumconic; specifically, an ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the line at infinity. Several well-known cubics (e.g., Thompson cubic, Darboux cubic, Neuberg cubic) are self-isogonal-conjugate, in the sense that if X is on the cubic, then X–1 is also on the cubic. (Wikipedia).
Highlights from triangle geometry (I) | WildTrig: Intro to Rational Trigonometry | N J Wildberger
The Euler line, the nine-point circle, incenters and isogonal conjugation. ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, stu
From playlist WildTrig: Intro to Rational Trigonometry
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
#MegaFavNumbers The Making of The Elements is a series in which I am documenting the process of my latest musical composition, which will fulfill my senior Honors project at Belmont University. This video is part of the MegaFavNumbers project, and documents the total number of combinatio
From playlist MegaFavNumbers
Mod-01 Lec-3ex Symmetry in Perfect Solids - Worked Examples
Condensed Matter Physics by Prof. G. Rangarajan, Department of Physics, IIT Madras. For more details on NPTEL visit http://nptel.iitm.ac.in
From playlist NPTEL: Condensed Matter Physics - CosmoLearning.com Physics Course
What is an equiangular triangle
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
Algebra can' t solve This Equation. But what can ?
Are you a visual thinker ? Today we explore a super fun problem which looks hopeless if you only apply algebra to it ! More on Fermat point: https://en.wikipedia.org/wiki/Fermat_point Explore 50 000 of triangle centres: https://faculty.evansville.edu/ck6/encyclopedia/etc.html Patreon: ht
From playlist Interesting math problems
This video defines and gives and example of isomorphic graphs. mathispower4u.com
From playlist Graph Theory (Discrete Math)
What is an equilateral triangle
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
Math 060 Fall 2017 111517C Orthonormal Bases, Orthogonal Matrices, and Method of Least Squares
Definition of orthogonal matrices. Example: rotation matrix. Properties: Q orthogonal if and only if its transpose is its inverse. Q orthogonal implies it is an isometry; that it is isogonal (preserves angles). Theorem: How to find, given a vector in an inner product space, the closest
From playlist Course 4: Linear Algebra (Fall 2017)
Isosceles & Equilateral Triangle Properties
I introduce 2 theorems about the properties of Isosceles and Equilateral Triangles. These theorems discuss how the base angles are congruent and that the bisector of the vertex is also a perpendicular bisector of the base. This video includes 2 proofs and 2 algebraic examples. EXAMPLES
From playlist Geometry
What is a perpendicular bisector
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
What are the names of different types of polygons based on the number of sides
👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
MIT 16.687 Private Pilot Ground School, IAP 2019 Instructor: Philip Greenspun, Tina Srivastava View the complete course: https://ocw.mit.edu/16-687IAP19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP63cUdAG3v311Vl72ozOiK25 This lecture introduced different aircraft sy
From playlist MIT 16.687 Private Pilot Ground School, IAP 2019
What is the difference between convex and concave
👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Complex Conjugate Root Theorem (2 of 2: Other conjugate properties)
More resources available at www.misterwootube.com
From playlist Using Complex Numbers
Complex Conjugate Root Theorem (Ch3 Pr17)
This problem shows that if a quadratic equation which has real coefficients, has a complex root alpha then so is the complex conjugate of alpha, and it is also true to higher degree polynomial.This is problem 17 in Chapter 3. Presented by Boris Lerner of the School of Mathematics and Stati
From playlist Mathematics 1A (Algebra)
Linear Algebra 5.3 Complex Vector Spaces
My notes are available at http://asherbroberts.com/ (so you can write along with me). Elementary Linear Algebra: Applications Version 12th Edition by Howard Anton, Chris Rorres, and Anton Kaul A. Roberts is supported in part by the grants NSF CAREER 1653602 and NSF DMS 2153803.
From playlist Linear Algebra
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
22. Acid-Base Equilibrium: Salt Solutions and Buffers
MIT 5.111 Principles of Chemical Science, Fall 2014 View the complete course: https://ocw.mit.edu/5-111F14 Instructor: Catherine Drennan A buffer helps to maintain a constant pH. Our blood has a natural buffering system to ensure that the pH of our blood stays within a narrow window and t
From playlist MIT 5.111 Principles of Chemical Science, Fall 2014