Question

The vertices of a triangle determine a circle, called the circumcircle of the triangle. Show that if P is any point on the circumcircle of a triangle, and X, Y, and Z are the feet of the perpendiculars from P to the sides of the triangle, then X, Y and Z are collinear.

Answer 1

In particular, consider point on the circumcircle of triangle as shown below. We want to find a point on each line containing segments , and that are nearest to . To determine the location these points, we construct three lines through that are perpendicular to the three sides of the triangle. The intersections of the lines and the sides, namely , , and as shown below are the points nearest to .

Notice that is extended since the point perpendicular to is not on the segment.

Some Elementary Observations

Before further discussion, we take note of the following observations:

1. Notice that we can show that points , , and are collinear if we can show that .
2. Looking at the figure above, we observe that is a cyclic quadrilateral.
3. Since , By Thales’ theorem, points , , and are on the circle. In addition, which also means that point is on the same circle as those of , and . In effect, is also a cyclic quadrilateral.
4. It follows from 3 that is also a cyclic quadrilateral.

We now use the observations above to prove our conjecture. We know that the measure of the opposite angles of a quadrilateral add up to . As we have mentioned above, proving that L, M, and N are collinear, it is sufficient to show that .

Proof

In quadrilateral , . Similarly, in quadrilateral , . It follows that . In addition, in quadrilateral ,

.

This is what we want to show.

The line above containing points , and is called the Simson line named after Robert Simson, a Scottish mathematician. The theorem we just proved above is the Simpson Line Theorem.