Triangle in a triangle: On a problem of Steinhaus

Rectangles in triangles

“…If this largest rectangle R max is λ max u by λ max v, then R itself fits in T if and only if λ max ≥ 1. Sullivan [8], extending a preliminary result of Post [6], has shown that if a convex polygon fits in a triangle in any way whatsoever, then it can be moved continuously within the triangle so that one side lies along a side of the triangle. Consequently this largest rectangle R max fits in T with one side along a side of T .…”

“…In 1993 Post [6], in answer to a question of Steinhaus [7], gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second, and Carver [1], solving a problem posed in 1956 by Ford [4], gave a necessary and sufficient condition on the four edge-lengths for one rectangle to fit into another (cf. Wetzel [9]).…”

Triangles in squares

“…Questions about precisely when one shape fits into another attract wide attention. In 1993 Post [1] gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second. Recently, the necessary and sufficient conditions for squares to fit in triangles [2], equilateral triangles in triangles [3], rectangles in triangles [4] and rectangles in rectangles [5] are given.…”

“…First we consider the circumscribed rectangles of triangle T such that the vertex A lies at one of the corners. Let the dashed rectangle shown in Figure 4 When T is inscribed in a rectangle with C lying at one of the corners (see the dashed rectangle in Figure 4 Then f(0), g(d) are always positive according to (1) and (2), and therefore only the function m (rj) has a value zero which implies that T has only one circumscribed square with the vertex C lying at the corner.…”

Minimum Area Isosceles Containers

“…As Steinhaus [21] pointed out, it is not even clear how to decide, whether a given triangle T can be brought into a position where it covers a fixed triangle T . The first such algorithm was found by Post [17] in 1993, and it was based on the following lemma. Lemma 1.1 (Post).…”