First the students observe that there are three adjacent triangles in this construction, each similar to each other. (Technically the triangles together form a fourth triangle) They will also notice that no matter the location of points c and d, point a can be adjusted so that c is a vertice of a triangle. So this construction can be useful for any length of PC and PD.
The questions are then meant to lead the students to use the ratios of the sides to determine a geometric mean. The lack of measurements in the applet is meant to discourage students from simply solving in terms of numbers instead of side lengths.
For example, the students might use PC:PB=PB:PA to determine that PB^2=PAPC and ultimately that PB=sqrt(PAPC). The construction offers ample practice for noticing other geometric means.
Throughout the prompts, the student is encouraged to explore other representations. This culminates in observing that the length of PB is the cube root of (PC^2)(PD). One way to solve this is to notice the 'two mean proportional' PC:PB=PB:PA=PA:PD. Students may also solve this by first solving that PB=2 if PC=1 and PD=8, and then do more exploration from there.
The student may then notice that if PD is 'a' times PC, then PB is 'the cube root of a' times PC. So, if PC is the side of a cube with volume B, then PB is the side length of a cube with volume aB. For example, if the length of PD is 2PC, then PB is the construction solution to 'cubing the square'.
