ATC Triangle Centers

Steve Phelps, Feb 14, 2017

Table of Contents

  1. Circles Associated with Triangles
    1. Constructing the Circumcircle
    2. Constructing the Incircle
    3. Constructing an Excircle
    4. Constructing the Nine-Point Circle
    5. The Circumcircle and the Nine-Point Circle
  2. Triangles Associated with Other Triangles
    1. The Cevian Triangle
    2. The Pedal Triangle
    3. The Wallace-Simson Line
    4. Minimizing the Triangle Perimeter
  3. Centers Associated with the Medial Triangle
    1. The Spieker Center
    2. The Nine-Point Center
    3. A Triangle Cleaver
    4. Cleavers and Angle Bisectors
    5. Three Cleavers Concur
  4. Centers Associated with the Incircle
    1. The Gergonne Point
    2. The Nagel Point
    3. The Feuerbach Point
  5. The Fermat Point
    1. Minimizing the Sum of Distances
    2. The Fermat Point Construction
  6. The Orthocenter
    1. Tracing the Orthocenter
    2. The Orthocenter Locus
    3. The Orthocentric Quartet

1.

Circles Associated with Triangles

  • 1. Constructing the Circumcircle
  • 2. Constructing the Incircle
  • 3. Constructing an Excircle
  • 4. Constructing the Nine-Point Circle
  • 5. The Circumcircle and the Nine-Point Circle

1.1.

Constructing the Circumcircle

– Steve Phelps

1.2.

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1.3.

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1.4.

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1.5.

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2.

Triangles Associated with Other Triangles

  • 1. The Cevian Triangle
  • 2. The Pedal Triangle
  • 3. The Wallace-Simson Line
  • 4. Minimizing the Triangle Perimeter

2.1.

The Cevian Triangle

For a given triangle ABC and a point P in the plane of the triangle, construct lines (cevians) AP, BP, and CP. Where these lines intersect the other sides, use these intersection points as vertices of a new triangle. This new triangle is the cevian triangle.
– Steve Phelps

2.2.

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2.3.

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2.4.

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3.

Centers Associated with the Medial Triangle

  • 1. The Spieker Center
  • 2. The Nine-Point Center
  • 3. A Triangle Cleaver
  • 4. Cleavers and Angle Bisectors
  • 5. Three Cleavers Concur

3.1.

The Spieker Center

The Spieker Center is the incenter of the medial triangle. It is the center of mass of the perimeter of the triangle.[br][br]Play through the construction, then try your hand at it below.
– Steve Phelps

3.2.

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3.3.

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3.4.

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3.5.

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4.

Centers Associated with the Incircle

  • 1. The Gergonne Point
  • 2. The Nagel Point
  • 3. The Feuerbach Point

4.1.

The Gergonne Point

Construct the incircle of a triangle. The lines drawn from the vertex to the opposite tangent point are concurrent.
– Steve Phelps

4.2.

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4.3.

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5.

The Fermat Point

  • 1. Minimizing the Sum of Distances
  • 2. The Fermat Point Construction

5.1.

Minimizing the Sum of Distances

Drag point P around the plane and try to make the sum of the distances from the vertices as small as possible.
– Steve Phelps

5.2.

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6.

The Orthocenter

  • 1. Tracing the Orthocenter
  • 2. The Orthocenter Locus
  • 3. The Orthocentric Quartet

6.1.

Tracing the Orthocenter

Drag point A along line DE. Notice the path that Orthocenter H traces. What is it, and how is it related to the other elements in the figure?
– Steve Phelps

6.2.

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6.3.

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