Given are two parallel lines a, b and two points on each: (A,A1) and (B,B1) respectively. For any real number x, construct the points Ax = (1-x)*A + x*A1, Bx = (1-x)*B + x*B1. The segment AxBx depends on x, and has the following properties. [1] AAx/AxA1 = BBx/BxB1 for every x. [2] Lines Lx = AxBx pass through the intersection point C of AB and A1B1. [3] AxBx/BxC is constant i.e. independent of x.
All properties are trivial consequences of Thale's theorem. The figure here is a preamble to the much more interesting figure, resulting by retaining all definitions except the parallelity of lines a and b. This is studied in LinesOfSegments2.html .
![[alogo]](alogo.png){kind=small}
Lines of segments ![[0_0]](LinesOfSegments_files/LinesOfSegments_0_0_0.png)
![[0_1]](LinesOfSegments_files/LinesOfSegments_0_0_1.png)
![[0_2]](LinesOfSegments_files/LinesOfSegments_0_0_2.png)
![[1_0]](LinesOfSegments_files/LinesOfSegments_0_1_0.png)
![[1_1]](LinesOfSegments_files/LinesOfSegments_0_1_1.png)
![[1_2]](LinesOfSegments_files/LinesOfSegments_0_1_2.png)