Geometric Constructions 11

  1. A wooden triangle moves on a plane so that two vertices move along two fixed perpendicular lines. How does the third vertex move?
  2. Generalized the above situation to two fixed lines not necessarily perpendicular to each other.
  3. Illustrate Napoleon's Theorem: the centers L, M, N of the three equilateral triangles DBXC, DCYA, DAZB built outwards on the sides BC, CA, AB of an arbitrary triangle DABC are the vertices of an equilateral triangle. The same is true of the centers of the three inward equilateral triangles.
  4. Generalize Napoleon's Theorem.
  5. Explore interesting properties associated with the Torricelli's configuration.
  6. Investigate the properties of the figure consisting of a quadrilateral together with the four squares erected outwardly on the sides.
  7. Design a figure to illustrate the following cases of indeterminate forms all at the same time: