A Comparison of an Orthic Triangle and a Triangle Formed by Extended Altitudes

Assignment #8, Problem #10

A Write-up by Ryan Fox

The problem, as it was posited on the website, was the following.

Examine the triangle formed by the points where the extended altitudes meet the circumcircle. How is it related to the Orthic triangle? Proof? Will the relationship still hold if the original triangle is obtuse?

The first part of this problem was to build a triangle.

From there we can construct the circumcenter and circumcircle by constructing the perpendicular bisectors, finding the intersection of those bisectors to determine the circumcenter and using that point and one of the vertices of the triangle to construct the circumcircle.

The next step would be to construct the three altitudes of the triangles and determine the intersection of the altitudes and the circumcircle. Connecting these three vertices would make up the desired triangle asked for in the problem. Triangle ABC, painted in red, is the triangle we are currently interested in.

From earlier investigations, we know that connecting the three feet of the altitude, the point that is the intersection of the altitude and the perpendicular side of the triangle, creates the orthic triangle. The construction of the orthic triangle, triangle DEF painted in brown, is given below.

We are interested to see if there is any relationship between triangles ABC and DEF. For the rest of this investigation, I will be referring to the triangles from in the construction in this Geometer’s Sketchpad file. As we can observe, angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to F. At this point, we are able to say that triangle ABC is at least similar to triangle DEF. It appears that the two triangles are merely similar and not congruent; determining the ratio between segment AB and segment DE provides verification for our suspicions. Now we want to see if the segments formed by the two triangles do indeed create two similar triangles, one twice as large as the other.

At this point, we want to prove our conjecture that the sides of the orthic triangle are half the length of the sides of the extended altitudes triangle. For an illustration of this proof, we will use the following picture, which can be accessed by the following Geometer’s Sketchpad file.

From the definition of orthic triangles, we know that D, E, and F all lie on the altitude of the original triangle, just like points A, B, and C. Since each vertex of the triangle is the vertex of an angle whose sides are the other two altitudes of the triangle, we can assume that the corresponding angles are congruent: angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F. If we look at the intersection of two of the altitudes, for example BX and AZ, there is a point of intersection, which we will label Q. By the reflexive property of congruence, we can say angle AQB is congruent to angle DQE. We can then construct the circumcircle for the orthic triangle. The center of the circumcircle of the orthic triangle is R and the center of the circumcircle of the extended altitudes triangle is J. RD is the radius of the circumcircle of the orthic triangle. If we construct a circle with center at J and radius of length equal to RD, the circle intersects the segment JA. By constructing a second radius of the circumcircle using the intersection point of JA, we see that the radius has its endpoint at A. From this we can say that JA is twice the length of RD. Using the same logic we can say that JB is twice the length of RE. We can say that angle AJB is congruent to angle DRE to show that the two triangles are similar. Since the ratio between the two sides of similar triangles is 2:1 we can say that AB is twice the length of DE. By the same logic, we can show that AC is twice the length of DF and BC is twice the length of EF. From this, we can say that the extended altitudes triangle is twice the orthic triangle.

Even when triangle XYZ becomes an obtuse, as can be made possible using the GSP file from the above link, it is possible that the two triangles maintain this relationship. In fact, the illustration in this write-up is of an obtuse triangle. In the link to this Geometer’s Sketchpad file, we will investigate the orthic triangle and the extended altitudes triangle for an acute triangle.