Pedal triangle

Abstract: It is standard in plane geometry to construct the three medians, angle bisectors, or altitudes of a given triangle T. The three lines of any set intersect the opposite sides (or their extensions) of the triangle in three points (the feet) that can be taken as the vertices of a new triangle T'. The process can be iterated. Of course the successive triangles become smaller, but we concentrate only on their angles. For example, the triangle formed from the feet of the medians is similar to the original triangle; the angles are unchanged. The successive triangles formed from the angle bisectors limit to an equi-angular triangle. On the other hand, the angles of the successive triangles formed from the feet of the altitudes behave in a much more complicated manner. In some sense, every type of behavior is possible. For any r, there are triangles whose angles repeat after r iterations (for example, a triangle with angles 360, 720, 720 is similar to its second iterate; a triangle with angles 120, 360, 1320 is similar to its fourth iterate). For any r and k, there are triangles whose angles repeat every r iterations after an initial delay of k iterations. For each k, there are triangles that after k iterations are equiangular (for example, 600, 105?, 15? with k = 2). There are lots of triangles whose angles never repeat (uncountably many). Indeed, there are triangles whose angles come arbitrarily close to any given triangle. There are triangles that become almost equiangular and stay that way for, say 1,000,000 iterations, but that then do something completely different. Hobson [3] called the triangle T' formed from the feet of the altitudes of a triangle T the pedal triangle. Recently Kingston and Synge [4] revisited and corrected Hobson's work and, for example, determined a criterion for some pedal iterate of T to have the same angles as T. The purpose of the present article is to revisit the issue again from a different mathematical point of view, which makes it routine to understand the behavior of the angles of successive pedal triangles. Let us consider the question of triangles whose angles repeat after r pedal iterations. Determining criteria directly on the angles leads to rather technical conditions [4] (and in fact is where Hobson made mistakes). A recurring theme in mathematics is to relabel objects so that desirable properties are more evident, and that is our present theme. For each triangle T, we assign a label E(T) that makes the behavior under the iterated pedal map obvious. Equally important, there is a straightforward way of determining the angles of the triangle from its label. More precisely, we consider all infinite sequences a1a2a3... of four symbols, say each ai equals 0, 1, 2, or 3. Every triangle T is labelled with one such sequence E(T), and two triangles are labelled with the same sequence if and only if they are

Abstract: In a recent article in the Monthly [2] Peter Lax proved a mixing property of the pedal mapping P which maps the set of angles of a triangle into the angles of the pedal triangle. His result tells us that if we start with almost any triangle, the sequence of successive pedal triangles contains triangles with shapes arbitrarily close to all possible shapes. In this note we represent the pedal mapping as a shift. We could have stopped there and referred to books on ergodic theory for proofs of mixing properties of shift mappings but we thought it would help many readers to give brief derivations of what is needed here, using only the elements of measure theory and probability theory. Lax follows Kingston and Synge [1] in representing a triangle of given shape as a point x in an equilateral triangle E = A PQR. The barycentric coordinates of x are the angles of the triangle in units of r. In this representation the pedal mapping is the result of the following operations. Connect the midpoints of the sides of E. Denote the four congruent triangles into which E is subdivided by EP, EQ, ERand EM where EP is adjacent to P etc. On the interior of a corner triangle the pedal mapping P is the dilatation by a factor 2 which maps the interior of the corner triangle onto the interior of E. On the interior of EM, P is a reflection in the centroid of E combined with a dilatation by 2. The boundaries of the small triangles correspond to right angled triangles and degenerate triangles for which the pedal triangle is degenerate and P is undefined. We want to consider iterates of P. The mapping P is not defined on the boundaries of the four small triangles. Its square is not defined on the preimages of these, which are the boundaries of the subdivisions of EP1,..., EM, etc. We get rid of these loose ends by restricting P to the set E' obtained by removing from E its boundary points and the boundary points of all triangles obtained by successive subdivisions. The removed po,ints form a set of measure 0. Let us choose units so that the measure of E' is 1, so that we can employ the language of probability theory. The mixing property we are going to prove is the following.

Abstract: This article presents interesting generalisations of three well-known results related to pedal triangles and distances, and the Simson line. The triangle whose vertices are the feet of the perpendiculars from a point P inside a triangle ABC to each of its sides AB, BC and AC, is called a pedal triangle.

Abstract: We generalize the construction of the ordinary Sierpinski triangle to obtain a two-parameter family of fractals we call Sierpinski pedal triangles. These fractals are obtained from a given triangle by recursively deleting the associated pedal triangles in a manner analogous to the construction of the ordinary Sierpinski triangle, but their fractal dimensions depend on the choice of the initial triangles. In this paper, we discuss the fractal dimensions of the Sierpinski pedal triangles and the related area ratio problem, and provide some computer-generated graphs of the fractals.