The Circles of Lester, Evans, Parry, and Generalizations

Steve Phelps, 22.01.2017

The Circles of Lester, Evans, Parry, and Their Generalizations From Paul Yiu's MathFest presentation in 2008

Inhaltsverzeichnis

  1. Basics
    1. Basic Triangle Centers
  2. Fermat Points
    1. Fermat Point and Negative Fermat Point
    2. Fermat Point and the Orthocentroidal Circle
    3. Fermat Points and the Centroid
    4. Fermat Points and the Orthocenter
    5. Circle Through the Fermat Points
  3. Kiepert Point
    1. The Kiepert Point (Perspector) and Kiepert Triangle
    2. Kiepert Hyperbola
    3. Points on Kiepert Hyperbola
    4. Circles Through Fermat Point & Kiepert Hyperbola
  4. The First Lester Circle
    1. The First Lester Circle
  5. The Symmedian and Isodynamic Points
    1. Inverses and Orthogonal Circle

1.

Basics

  • 1. Basic Triangle Centers

1.1.

Basic Triangle Centers

From Paul Yiu's MathFest presentation in 2008[br]
Basic Triangle Centers, Circles, and Lines.
Circumcircle, nine-point circle, the Euler line, the circumcenter, centroid, nine-point center, and the orthocenter.[br][br]HN : NG : GO = 3 : 1 : 2
The Orthocentroidal Circle
N and O are inverses with respect to the orthocentroidal circle.
Euler Line Reflections
Reflect the Euler line in each side of the triangle. The reflections concur at a point on the circumcircle.
The Fermat Point
Construct equilateral triangles on the exterior of the triangle. The lines AA', BB', and CC' are concurrent. This point is the Fermat point.
The Negative Fermat Point
Construct equilateral triangles on the interior of the triangle. The lines AA', BB', and CC' are concurrent. This point is the Negative Fermat point.
– Steve Phelps

2.

Fermat Points

  • 1. Fermat Point and Negative Fermat Point
  • 2. Fermat Point and the Orthocentroidal Circle
  • 3. Fermat Points and the Centroid
  • 4. Fermat Points and the Orthocenter
  • 5. Circle Through the Fermat Points

2.1.

Fermat Point and Negative Fermat Point

Building equilateral triangle on the exterior of the triangle leads to the Fermat Point (Left)[br]Building equilateral triangle on the inxterior of the triangle leads to the Negative Fermat Point (Right)
– Steve Phelps

2.2.

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2.3.

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2.4.

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2.5.

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3.

Kiepert Point

  • 1. The Kiepert Point (Perspector) and Kiepert Triangle
  • 2. Kiepert Hyperbola
  • 3. Points on Kiepert Hyperbola
  • 4. Circles Through Fermat Point & Kiepert Hyperbola

3.1.

The Kiepert Point (Perspector) and Kiepert Triangle

Construct isosceles triangles with congruent base angles on each side of the triangle. The lines drawn from the vertex of each isosceles triangle to the opposite vertex AX, BY, and CZ are concurrent at the Kiepert Point. [br][br]The three isosceles triangle vertex points form a triangle called the Kiepert Triangle.[br][br]Triangles ABC and XYZ are in perspective, and the Kiepert point is also referred to as the Kiepert Perspector.
– Steve Phelps

3.2.

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3.3.

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3.4.

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4.

The First Lester Circle

  • 1. The First Lester Circle

4.1.

The First Lester Circle

The Fermat Point and the Negative Fermat Point are concyclic with the Nine-point center and the Circumcenter.
– Steve Phelps

5.

The Symmedian and Isodynamic Points

  • 1. Inverses and Orthogonal Circle

5.1.

Inverses and Orthogonal Circle

Points D and E are inverses in the circle O with diameter CB. Every circle through D and E is orthogonal to circle O. [br][br](Drag point D, too.)
– Steve Phelps

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