Assignment Due Wednesday 1/23

 

  1. (One last time). Given a circle with center O and radius r and given points A and B not collinear with O, prove that triangle OAB is similar to triangle OB'A'.
  2. Given two concentric circles c1 and c2, with radii r1 and r2. Let S1 denote the transformation inversion in c1 and S2 denote inversion in c2. Then the composition S2S1 is a transformation of the I-plane. First, given a point P, describe exactly what point is Q = S2S1(P). Next tell what kind of familiar transformation S2S1 is.
  3. Draw a triangle ABC and construct the circle c inscribed in the triangle. The construct the image of triangle ABC by inversion in c. (To be precise, construct the images of the 3 segments AB, BC, CA.)
  4. Construct a square ABCD inscribed in a circle c (or a circle c circumscribed around a square ABCD for the same starting figure). Construct the image of the sides of the square by inversion in c.
  5. (Sketch and talk). In this question, tell what the construction would do and sketch it, but you do not have to carry out the construction precisely.

        Suppose two circles c1 and c2 do not intersect and that the circles d1 and d2 are orthogonal to both c1 and c2. Then d1 and d2 intersect in two points A and B (you should know why). Suppose that you invert all 4 circles in a circle e with center A that passes through B. Sketch and describe what inverted figure looks like, pointing out key relationships.

        Suppose that a figure consists of two parallel E-lines m1 and m2. Also suppose that n is an I-circle orthogonal to both m1 and m2. Tell and sketch what this figure must look like. Then sketch and tell what the inverted figure looks like if you invert in a circle with center on an I-point P not on any of m1, m2, or n. Then make a new sketch and tell what the inverted figure looks like if you invert in an I-point Q that is on n but not on m1 or m2.

        Sketch a figure consisting of two circles c1 and c2 tangent at a point A and also a third circle c3 exterior to the other two. Then sketch the inversion of this figure in a circle centered at A (any circle will do for a sketch, why?)


  1. In the figure below, construct a circle d1 that is tangent to c1 and c2. Then construct a circle d2 which is tangent to all 3 circles c1, c2, d1.