Geometric Constructions 5

  1. Show that :
    1. there are three parallel tangents to the cardioid with any given gradient; if we connect the points of tangency to the cusp, the three segments meet at equal angles of 2p/3; the area of the triangle formed by the points of tangency is constant;
    2. the tangents at the ends of any chord through the cusp of a cardioid are at right angles;
    3. the length of any chord through the cusp of a cardioid is constant;
  2. Construct the rectangle enclosed by the tangents and normal at the ends of any chord through the cusp of a cardioid. What's so particular about the vertices of this rectangle?    (JavaSketchpad file)
  3. Take five points on a cardioid with the corresponding points on the base circle forming a regular pentagon. What properties does the figure have?
  4. Construct an animation displaying a cardioid sliding along two orthogonal straight lines.
  5. Illustrate the principle behind the cardioid condenser.
  6. Illustrate the harmonic motion associated with the rotation of the cardioid about its cusp.
  7. Construct the formation of the envelope of light rays, emitted from a radiant point source on a circle, after reflection by the same circle.
  8. Construct an animation displaying two cardioids having the same cusp and are orthogonal to each other.