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.