Altitudes and Circumcircles

Let ΔABC be any acute triangle, and label it as follows (see below for a diagram): Draw the three altitudes of ΔABC, and let D, E, and F be the intersections of these altitudes with the side of the triangle to which they are perpendicular. Finally, let P, Q, and R be the intersections of the altitudes with the circumcircle (other than A, B, and C).
Using this GSP file, make a conjecture about the following sum:
Can you find any pairs of congruent segments?
It looks like Let's prove it!
First, let's modify the sum a little bit.
So if we want to show that ,
Let's try the second one.
Draw in hexagon CPBRAQ and label some angles as shown:
We know that two angles on a single circle which subtend the same segment are congruent (see Assignment 9 for a "proof"). and both subtend segment , so they must be congruent. Thus, c = a + b. (1)
Since and are both right,
Furthermore, since is a straight angle, we have
By (1) and (2), we have
d = a + b = c
Also, , since both are right angles. And finally, , clearly.
So by AAS congruence, we have
Thus,
By similar reasoning, we have
Because of the congruences we just found, we can rewrite (*) as:
Let's look at a few areas.
For the whole triangle, ΔABC, we have
Now let's break the area up into three pieces:
We now have:
So we have
This is precisely what we wanted to show (see (**))! Q.E.D.
