Seven Functions on a Triangle
- Judah L Schwartz
Draw the three medians in a triangle. The three points at which the medians intercept the sides form a new triangle. Similarly, for the angle bisectors and for altitudes (in non-obtuse triangles). This means that you can think of each of these constructions as a function from a domain of triples of points in the plane to a range of triples of points in the plane. You can use this environment to explore the relationship between the original triangle and the one formed by the intersections of the medians, angle bisectors and altitudes with the sides of the triangle. Here is another set of functions on a triangles - It is well known that the angle bisectors of the angles of a triangle all meet at a point, as do the altitudes and the medians. In each case reflect the meeting point over the three sides of the triangle - the resulting three points form a new triangle. You can use this environment to explore the relationship between the original triangle and the one formed by reflecting the three, in general different, points of concurrence over the sides of the triangle. Finally, you can reflect each of the vertices of the triangles over the side opposite it. Once again a new triangle is formed. Drag the points A, B, and C to make the triangle you want to examine. What conjectures can you make about these seven functions? Do these functions have inverses? What happen when you compose these functions with themselves? with one another? What conjectures would you like to explore? Which conjectures do you think might be easy to prove?