Thales, a Greek mathematician, was the first to formulate the Fundamental Proportionality Theorem. As a result, it is sometimes referred to as the Thales Theorem. Thales was the first to initiate and establish the Theoretical Study of Geometry, which he intended to use to make astronomy a more precise science.

Proportionality at its most fundamental level A line parallel to a side of a triangle that crosses two additional sides into two different points violates the BPT, which asserts that if the line divides those sides in proportion, then the line divides the BPT. 

Basic proportionality theorem 

If two intersecting lines are intercepted by a pair of parallels, then it is known as the intercept theorem which is also known as Thales’s theorem, basic proportionality theorem, or side splitter theorem, is an important theorem in elementary geometry that describes the ratios of various line segments that are created when two intersecting lines are intersected by a pair of parallels.

If two triangles are similar to one another, then they are said to be similar.

1. The angles of the corresponding triangles of both triangles are equal.

2.  The sides of both triangles that correspond to each other are in proportion to one another.

Proof of Basic proportionality theorem 

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Let us take ∆ABC where DE||BC 

We have to prove AD/DB = AE/EC

Now, join BE and CD

Draw DM ⊥ AC and EN AB

Area of ADE = 1/2 x base x height

= ½ x AD x EN ……..1

Area of BDE = ½ x Base x height 

= ½ x DB x EN …….2

Divide 1 and 2

Area of ADE / Area of BDE = ½ x AD x EN / ½ x DB x EN 

Area of ADE / Area of BDE = AD/DB ……………. (A)

Area of ADE = 1/2 x base x height

= ½ x AE x DM ……..3

Area of DEC = ½ x Base x height 

= ½ x EC x DM …….4

Divide 1 and 2

Area of ADE / Area of DEC = ½ x AE x DM / ½ x EC x DM

Area of ADE / Area of DEC = AE/EC ……………. (B)

Now, 

AD / DB = AE / EC (from A and B) 

Hence proved.

Converse of basic proportionality theorem   

Let us take a triangle ABC, in which DE intersects the sides AB, AC at D and E, such that AD/DB = AE/ EC.

Let us draw some lines DE’ which is parallel to BC. We have to prove DE and DE’ are the same line or the points E and E’ coincide. 

Given: AD/DB = AE/EC …….(1)

Here, DE’ || BC (drawn) 

By BPT

AD/DB = AE’ / CE’ ……(2)

From 1 and 2 we get,

AC/EC = AE’ / CE’ 

AE/EC = AE’/CE’

AE/EC +1 = AE’/ CE’ + 1 

AE + EC / EC = AE’ + CE’ / CE’ 

AC/EC = AC / CE’ 

1/EC = 1/CE’ 

EC = CE’

Conclusion 

Thales’s Theorem, also known as Thales’s intersectionality theory, is an important theorem in elementary geometry that describes the ratios of various line segments that are created when two intersecting lines are intersected by a pair of parallels. As a result, it is sometimes referred to as the Thales Theorem.