Interactive Geometry Dictionary
Properties of the Simson
Line How is the angle between two different Simson
lines related to the arc between the two points?
The applet below shows two points P and P' on the circumcircle
of a triangle and the Simson lines that belong to them. The measure of
the arc between points P and P' and the measure of the angle
between their Simson lines are given. Drag either of the points or a vertex
of the triangle to change the diagram. What is the relationship between
the arc and the angle?
Theorem
The measure of the angle
between the Simson lines belonging to two points P and P'
is half the measure of the arc P'P.
How is the Simson line of a triangle related
to the orthocenter of the triangle? The applet below shows the intersection S of the Simson line
corresponding to point P with segment HP that connects the
orthocenter H to the point P. You can drag the vertices
of the triangle to change its shape and you can drag point P to
change its location on the circumcircle of the triangle. Use the given
measurements to see if you can determine the relationship between the
segment HP and the Simson line corresponding to point P.
Theorem
The Simson line
for the point P on the circumcircle of a triangle bisects segment
HP joining the point P to the orthocenter H of the
triangle.
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Pre-K - 2 
