Task

1 Adding scenes

2     Incremental scenes

 The incremental scenes in this construction show the Euclidean construction of the bisector of the segment AB.

Scene 0. Bisector of the segment AB.

Scene 1. Segment AB.

Scene 2. Circle k(B,A).

Scene 3. Circle k(A,B).

Scene 4. C,D are the intersections of circles k(A,B) and k(B,A).

Scene 5. The line CD is the bisector of AB.

3     Differential scenes

 The differential scenes in this figure show some Bankoff circles of an arbellos.

Scene 0. An arbelos.

Scene 1. A basic property of arbellos.

Scene 2. Archimedes' twins in an arbellos.

Scene 3. The 3rd Bankoff circle.

Scene 4. The 4th Bankoff circle.

Scene 5. The 7th Bankofff circle.

4     Observed properties

 The scenes of this construction illustrate some properties of the following configuration: ABCD is a square, the line CE if tangent at E to the circle with diameter AD.

Scene 0. The configuration.

Scene 1. Similar triangles △ABE and △DAE

Scene 2. Area(ABE) = Area(ACE)

Scene 3. The circle through A,E,B is tangent to the line DB.

Scene 4. Common intersection: line AC, line BE, circle k(D,E,C).

Scene 5. |AE| : |EC| = |EB| : |BD|

Scene 6. ∠AC,BE = ∠DE,DC = ∠EC,ED