Task D

Take any triangle ABC. Construct isosceles triangles externally on each side. Let A', B' and C' be the external vertices of the externally constructed isosceles triangles with A' off the side opposite vetrex A, etc.

Construct lines AA', BB' and CC'. Observe. Are they concurrent? Is the point of concurrency any of the orthocenter, centroid, incenter or circumcenter of triangle ABC? If not what can you find out about this point.

Rather than limit the investigation to the case of isosceles triangles whose height is equal to their base let us explore the full range of isosceles triangles - that is we will consider isosceles triangles whose base angles are equal (similar isosceles triangles).

Some reflection will suggest that we are dealing with a general case of the other tasks discussed:


1. When the base angles are 60deg the exterior triangles are equilateral and we have Task C and the point of concurrency of the lines AA', BB' and CC' is the Fermat point of the triangle.

2. When the exterior triangles have base angles of 30deg we have Task A, when the base angles are 45deg we have Task B and when the base angles are -60deg (or equilateral triangles built toward the inside) we have Task E.

3. When the base angles are 0deg - it is quite obvious that the point of concurrency of the lines AA', BB' and CC' is the centroid

It seems reasonable to expect that the position of D will be determined by the base angle of the externally constructed isosceles triangle. We therefore expect a locus for D that will incorporate the five points of concurrency shown. The figure below illustrates the locus of D as the base angle of the isosceles triangles vary from 90deg to -90deg.

Our predictions are confirmed. What about the other points that we are familiar with: Orthocenter (H), Circumcenter (C), Incenter (I) and center of the nine point circle (N)?

It seems as though the Orthocenter is indeed a member of the locus while the Circumcenter (C), Incenter (I) and center of the nine point circle (N) do not lie on the locus. As with the Centroid it makes sense that the Orthocenter (H) should lie on the the locus - as the base angle tends toward 90deg so the intersection of the lines AA', BB' and CC' tend toward the Orthocenter.

For a GSP sketch that draws the locus of point D click here.

More than answering questions our discovery of a locus that looks like a hyperbola begs further questions, for example:

How does the shape of triangle ABC influence the shape of the locus? Let us consider the following cases (in the obtuse-angled cases we will make the obtuse angle greater than 120deg so that the fermat point is outside the triangle) click on the case to view a sketch:

1. ABC is an equilateral triangle.
2. ABC is an acute-angled isosceles triangle.
3. ABC is a right-angled isosceles triangle.
4. ABC is an obtuse-angled isosceles triangle.
5. ABC is an acute-angled scalene triangle.
6. ABC is a right-angled scalene triangle.
7. ABC is an obtuse-angled scalene triangle.

Summary of observations made:

1. If ABC is an equilateral triangle the locus is a point!
2. In all cases where triangle ABC is isosceles the locus is a a straight line passing through the vertical angle and bisecting the opposite side.
3. In all cases where the triangle ABC is scalene the locus is a hyperbola one arm of which passes though the two vertices at the largest and smallest angles of triangle ABC and the other arm touching the triangle ABC at the vertex that is at the smallest angle.
4. For each of our points there is a critical for which that point will coincide with a vertex of the triangle.