Monday, January 17, 2011

Triangle Centers, Simson Lines, Steiner Deltoid

Here are my Grasshopper methods to find some specific centers for a given triangle.

With the help of these centers, it's possible to draw Simson Lines and a Steiner Deltoid in Grasshopper.

The envelope of Simson Lines is the Steiner Deltoid.

Another way to build the deltoid is to draw two significant circles for a given triangle:
1.Feuerbach Circle(Nine-Point Circle): The circle that passes through nine significant points in a triangle

 Feuerbach Circle(Nine-Point Circle)
  • The midpoint of each side of the triangle
  • The foot of each altitude
  • The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).
     2. Steiner Circle: The circle whose center is the center of the Feuerbach Circle and whose radius is 1.5 times bigger than the circumcircle of the triangle.

When the Feuerbach Circle rolls around inside the Steiner Circle, a point on it traces the Steiner Deltoid.

Steiner Deltoid from caglar on Vimeo.

Definition has a method to find the Fermat Point of a triangle as well.
ghx file

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