Assignment # 9

Fall 2002 Semester

GSP - Pedal Triangles

Pedal Triangles

Observation 9.11  Envelope of the Simson Line

  Observations of the following items are shown below:

1.  Traces of Pedal point around the circumcircle, circle with radius greater than the circumcircle and incircle.

2.  Within each of the circle traces, include changes to the original triangle for equilateral triangle, isosceles triangle and obtuse angle triangle.


Conclusions: The envelope of the Simson line of the triangle ABC is the deltoid area (tricuspoid) displayed in the attached sketches.   The area of the deltoid is one half the area of the circumcircle. 

Also,  each side of the triangle is tangent to the deltoid at a point whose distance from the midpoint of the side equals the chord of the 9- Point Circle cut off by that side.


Circumcircle Cases:

Equilateral Triangle:  The sketch below shows the deltoid created by the traces of the line traced by the Simson line as the Petal point is rotated around the circumcircle.  Essentially, the tangent point of the deltoid area is at the midpoint of each side. 

When the triangle consists of 2 acute and one obtuse angle, then the traced area now lines up with the nine-point circle, as shown…. 

Note that for an isosceles triangle, we’ll end up with the tangent being at the midpoint on one of the sides and the other two congruent sides will have the same relative tangential point to the midpoint of the side.


Circle with radius greater than the circumcircle

When we trace the Simson line with the Petal point traveling on a circle slightly larger than the circumcircle, then the pattern resembles the pattern we showed with the Petal point traveling along the circumcircle.  In this case, the tangent points with the sides disappear and the shape is no longer the tricuspoid we saw with the circumcircle.  The sketch below is when there were 3 acute angles in the original triangle.

The diagram below shows the sketch of when the triangle is equilateral.  The shape is the same as in the 3 acute angles.  There is less area of this trace within the triangle as for the first case.  Also note that the vertices are the limits for the trace area. 


For the isosceles triangle, more of the triangle area is covered by the  trace.  The congruent sides resemble each other in this case. 


Incircle traces for the cases of isosceles shown in the sketch below.  This more resembles the case of the trace about the circumcircle.  More of the circumcircle area is covered by this trace than in the circumcircle trace above.

The next sketch shows the same type of trace for an equilateral triangle.  Note how the blank trace area is more consistent between the vertices, as would be expected. 


The last triangle is a general triangle with an obtuse angle.  As would be expected, the area of no traces is greater where the angles are more acute.  The no trace area is much less as the angle becomes more obtuse.