Constructing the Simson Line of a Triangle (Interactively!)

You can construct a Simson line for a triangle using the steps shown in the interactive math tool below.

• Step 1. Extend the sides of the triangle.
• Step 2. Construct the circumcircle of the triangle.
• Step 3. Construct an arbitrary point P on the circumference of the circumcircle. (Arbitrary means the point P can be anywhere on the circle.)
• Step 4. Drop perpendiculars from point P to the three sides of the triangle.
• Step 5. Construct the line that passes through the three feet P(1), P(2), and P(3) of the perpendiculars constructed in Step 4. This is the Simson line.

You can drag the vertices of the triangle to change its size or shape or drag point P to change its location on the circumcircle.

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Definition

The Simson line of a triangle belonging to the point P on the circumcircle of a triangle is the line passing through the feet of the perpendiculars dropped from the three vertices of the triangle to the opposite sides. William Wallace discovered the Simson line in 1799, but it was named after the British mathematician Robert Simson.

A Closer Look at the Definition of the Simson Line — Why Is It That Way?

Why must the point P used to define the Simson line of a triangle be on the circumcircle of the triangle? Look at the following sketch, in which three perpendiculars have been dropped to the sides of a triangle from a point Q that can be located anywhere. When are the three points Q(1), Q(2), and Q(3) collinear?

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Theorem

The three feet or perpendiculars dropped from a point Q to the sides of a triangle are collinear only when the point Q is on the circumcircle of the triangle.

What are Some Properties of the Simson Line?

• Find out how two different Simson lines for the same triangle are related.
• Find out how the Simson line of a triangle is related to the orthocenter of the triangle.

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References and Credits