Question

In the given diagram, which of the following statements is NOT true:

AB/BC=DE/EF


GB/GA=GE/GD


AB/EF=DE/BC


AB/DE=AC/DF

Answer 1

GB/GA=GE/GD

Explanation:

Answer 2

The third statement
`((AB)/(EF)=(DE)/(BC))` is NOT TRUE.

Explanation:

The Thales' theorem says that if two intercepting lines are intercepted by a group of parallel lines:

1. The ratio of any two segments on the first line equals the ratios of the according segments on the second line.
2. The ratio of the two segments on the same line starting at G equals the ratio of the segments on the parallels.

The first statement in the question can be proved using the first theorem's statement:

`(AB)/(BC) = (DE)/(EF)`

The second statement in the question can be proved using the second theorem's statement:

`(GB)/(GA) = (GE)/(GD)`

The third statement in the question can be proved FALSE using the first theorem's statement, because the ratios are being stablished between segments that are NOT corresponded.

`(AB)/(EF) = (DE)/(BC)` would mean that
`(AB)/(DE) = (EF)/(BC)`

But aplying the theorem we know that
`(AB)/(DE) = (BC)/(EF)` and both statements CAN'T BE TRUE for every resulting segment.

The fourth statement in the question can be proved true rearranging the equation.

`(AB)/(DE) =(AC)/(DF) \Rightarrow (AB)/(AC) =(DE)/(DF)`

This way, it becomes clear that the ratio is being established between corresponding segments in each line, just like the Thales' theorem says.

In conclusion, the third statement is the one that is NOT TRUE.