Modern Euclidean Concepts and Theorems

canhalt_1, Julie Buasen, Sep 18, 2020

This chapter considers some fundamental concepts and theorems concerning polygons (particularly triangles) and circles – concepts and theorems that have long been a part of Euclidean Geometry. It comprises of the special line segments in a triangle, the Euler Line and the Nine-point circle, and the excenters and the nine-point circle.

Table of Contents

  1. Secondary Parts of a Triangle
    1. Circumcenter
    2. Incenter
    3. Orthocenter
    4. Centroid
  2. The Euler Line and the Nine-Point Circle
    1. The Euler Line and the Nine Point Circle
  3. The Excenters and the Orthic Triangle
    1. The Excenters and the Orthic Triangle

1.

Secondary Parts of a Triangle

This section allows you to explore the different segments in a given triangle and their points of concurrency

  • 1. Circumcenter
  • 2. Incenter
  • 3. Orthocenter
  • 4. Centroid

1.1.

Circumcenter

All three [b]perpendicular bisectors[/b] of the sides of a triangle will intersect at the same point - the [b]circumcenter[/b].

[br][br]Move the points of the triangle to see how the circumcenter changes.
How does the circumcenter change as the triangle changes? What can you predict about the location of the circumcenter based on the type of triangle?

[br][br]How does the Perpendicular Bisector Theorem, and its converse, help us show that the perpendicular bisectors of a triangle must intersect at the same point?

[br][br]The circumcenter is the center of the circle that circumscribes the triangle.

How would you describe, in words, the length of the radius of the circle that circumscribes a triangle?
– canhalt_1

1.2.

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1.3.

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1.4.

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2.

The Euler Line and the Nine-Point Circle

The orthocenter,the centroid and the circumcenter of a non-equilateral triangle are aligned; that is to say, they belong to the same straight line, called line of Euler

  • 1. The Euler Line and the Nine Point Circle

2.1.

The Euler Line and the Nine Point Circle

The orthocenter,the centroid and the circumcenter of a non-equilateral triangle are aligned; that is to say, they belong to the same straight line, called line of Euler. In any triangle, the mid-points of its sides, the feet of its altitudes, and the mid
1. A triangle has how many, nine-point circles? Euler lines?[br][br]2. In general, a triangle and its nine-point circle have how many points in common?[br][br]3. Give the maximum and minimum number of points a triangle and its nine-point circle might have in common?[br]
– Julie Buasen

3.

The Excenters and the Orthic Triangle

  • 1. The Excenters and the Orthic Triangle

3.1.

The Excenters and the Orthic Triangle

The excenter is the intersection of the bisectors of two exterior angles and that of the remaining interior angle. The excenter is the center of the excircle, the circle that is tangent to extensions of two sides and the remaining side.
The excenter is the intersection of the bisectors of two exterior angles and that of the remaining interior angle.  The excenter is the center of the excircle, the circle that is tangent to extensions of two sides and the remaining side. [br][br][b]Consider the following:[/b][br][list=a][*]The [b]altitudes and sides of triangle GHD are interior and exterior angle bisectors of orthic triangle ABC[/b], so K is the incenter of triangle ABC and G, H, D are the 3 excenters [/*][br][*]The [b]sides of the orthic triangle form an "optical" or "billiard" path[/b] reflecting off the sides of ABC.[/*] [*]The orthic triangle [b]ABC[/b] has the [b]smallest perimeter[/b] of any triangle with vertices on the sides of [b]GHD[/b].[/*] [*]The altitudes and sides of triangle [b]GHD [/b]are interior and exterior angle bisectors of orthic triangle [b]ABC[/b], so K is the incenter of triangle [b]ABC[/b] and G, H, D are the 3 excenters. [/*][/list][br][b]**PROVE THAT THE ALTITUDES ARE ANGLE BISECTORS OF THE ORTHIC TRIANGLE.[/b][br]
– Julie Buasen

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