## The Simson Line

Given a triangle ABC and its Circumcircle. Select a point P on the circumcircle. From P, construct perpendiculars to the lines of the sides of the triangle with the feet of the perpendiculars determining intersetion points X, Y, and Z. **Show that X, Y, and Z are collinear.**

**GSP File for this construction**.

1. Explore different locations for Point P. What if it is at a vertex?

2. Contruct a point Q on the opposite end of a diameter from P. Locate the three points that are the feet of the perpendiculars from Q and determine the line. **Prove that the P-line and the Q-line are perpendicular.**

3. Trace the locus of the intersection of the P-line and the Q-line as point P (and Q) moves around the circumcircle. **Prove that the locus is the Nine Point Circle.**

4. Examine the envelope of lines formed by the line containing X, Y, and Z as point P is moved around the circumcircle.

**Who was Simson?**