1. Given a triangle ABC, construct a similar triangle A'B'C' so that the area of A'B'C' is twice the area of ABC.

2. Use GSP to explore the following situation.   Take four random points in the plane,  A, B, C, D,  and form the segments AB, BC,CD and DA.    Make a quadrilateral by connecting the midpoints of the four segments.   What do you notice about this quadrilateral?    State your observation in the form of a theorem,  and prove the theorem!

The next set of problems deals with the famous "Golden Ratio".

3. Construction:

Step one:  Construct a right triangle ABC,  with right angle at B,   such that the ratio of sides AB and BC is  2:1.

Step two:  Let D be the point where the circle through B with center C meets the hypotenuse.

Step three:  Let E be the point where the circle through D with center A meets AB.

The ratio  |BE| : |EA| is called the Golden Ratio.

4. Algebra:

a) Work out the exact value of the golden ratio in the following way:
Set |BC|=1 and |AB|=2.  Use the pythagorean theorem to get |CA|,  and then do the algebra to get the value of the golden ratio.

b) Show that  |EA| : |AB| is also the Golden Ratio.   In other words,  a  point placed on a line segment cuts the segment in the golden ratio if the ratio of the smaller to the larger equals the ratio of the larger to the whole.

5. Theorem:

Now prove the following theorem:   Given a rectangle  R whose sides are in the golden ratio,  draw a perpendicular through the longer side in such a way that R is divided into two rectangles R1  and R2,  such that R1 is a square.   Then R2 is  similar to R.