Finished product　　Extended problem-1　　Extended problem-2

A Quadrilateral Projected into a Square

　The main idea : given a quadrilateral, we can find a vanishing point such that this quadrilateral projected into a square. Sometimes ideas like this are useful on a stage. The light in a stage is improtant, since the shadows of anything on the stage should be engraved precisely. When a square shadow is needed, this idea can work ! It shows how to make it from finding the location of the light with some given quadrilateral.

　Now, we'll make it in 7 steps in Cabri 3D.

Step 1: Make a quadrilateral ABCD.

Step 2: Make lines AC, BD, the intersection is O.
Make lines AB, CD, the intersection is E, and suppose E is on BC side.
Make lines AD, BC, the intersection is F, and suppose F is on AB side.
Make lines EO and FO.

Step 3: Make line L1 which pass D and parallel to line EF.
Line L intersects line EO, FO and BD at G, H and I, respectly.

Step 4: Make the midpoint of G and H, called M.
Make the midpoint of C and I, called N.

Step 5: Make a sphere centered at M with radius GM.
Make a sphere centered at N with radius IN.
The intersection of these two spheres is a circle, called Q.

Step 6: Take a point J on the circle Q.
Take a plane containing J and line L1, called K.
(May make an animation on point J.)
Make a square centered at J on plane K with a vertex D.
Lable this square by A'B'C'D.

Step 7: Make lines AA', BB' and CC', these three lines intersects at one point.
Call this intersection P.
Make line DP.
Then "P - ABCD - A'B'C'D" is desired.