Geometric Construction 6

1. Construct the conic given two tangents, the corresponding points of tangency, and another point of the curve.
   
   

   1. Construct the lines and
      
      

   2. Find the intersection of the lines and From draw a variable ``Pascal line''.
      
      

   3. Locate the points and on the Pascal line.
      
      

   4. The required ``sixth point'' is constructed by taking the intersection of the lines and
      
      

2. Show that the deltoid can be generated as a locus in two ways.
   
   

   1. Construct the deltoid according to definition.
      
      

   2. Draw a straight line joining the point of tangency and the point on the curve.
      
      

   3. From the new intersection of the straight line with the circle construct the diameter.
      
      

   4. Draw the tangent to the deltoid.
      
      

   5. The required circle is the circumcircle of the triangle formed by the three straight lines.
      
      

3. Show that the astroid can be generated as a locus in two ways.
   
   

4. Show that the cardioid can be generated as a locus in two ways.
   
   

5. Show that the nephroid can be generated as a locus in two ways.
   
   

6. Construct circles which envelope both a hypocycloid and an epicycloid.