Basic Proportionality Theorem Proof

Let’s prove the Basic Proportionality Theorem.

Given: Let us suppose we have a triangle ABC, if we draw a line LM parallel to side BC.

To Proof: [Tex]\frac{AL}{LB}=\frac{AM}{MC}               [/Tex] 

Construction

To prove the required result, construct the following lines in the given figure:

• Construct LY⊥AM.
• Construct MX⊥AL.
• Join points L with C and M with B with a line segment.

Basic Proportionality Theorem Proof

Since, area of triangle= [Tex](\frac{1}{2}\times base\times height) [/Tex]

Area of [Tex]\triangle[/Tex] ALM=[Tex](\frac{1}{2}\times AL\times MX) [/Tex]

Area of [Tex]\triangle[/Tex] LBM=[Tex](\frac{1}{2}\times LB\times MX) [/Tex]

Area of [Tex]\triangle[/Tex] ALM=[Tex](\frac{1}{2}\times AM\times LY) [/Tex]

Area of [Tex]\triangle[/Tex] LMC=[Tex](\frac{1}{2}\times MC\times LY) [/Tex]

Ratio of area of [Tex]\triangle[/Tex] ALM and [Tex]\triangle[/Tex] LBM:

[Tex]\frac{area(\triangle ALM)}{area(\triangle LBM)}=\frac{(\frac{1}{2}\times AL\times MX)}{(\frac{1}{2}\times LB\times MX)}=\frac{AL}{LB}[/Tex]. . .(1)

Ratio of area of [Tex]\triangle[/Tex] ALM and  LMC:

[Tex]\frac{area(\triangle ALM)}{area(\triangle LMC)}=\frac{(\frac{1}{2}\times AM\times LY)}{(\frac{1}{2}\times MC\times LY)}=\frac{AM}{MC}[/Tex]. . .(2)

According to the property of triangles, the triangles on the same base and between same parallel lines have equal areas.

Therefore, [Tex]\triangle[/Tex] LBM and [Tex]\triangle[/Tex] LMC have equal areas. 

i.e., area of [Tex]\triangle[/Tex]  LBM = area of [Tex]\triangle[/Tex] LMC. . .(3)

From equations (1),(2), and (3) we can conclude:

[Tex]\bold{\frac{AL}{LB}=\frac{AM}{MC}}[/Tex]  [Hence Proved]

Corollary of Thales Theorem

Mid Point theorem states that, “If the line is drawn from one of the midpoints of the side of a triangle parallel to the second, then it always intersects the third side at the midpoint as well.”

Articles related to Basic Proportionality Theorem:

• Mid Point theorem
• Figures on the Same Base and between the Same Parallels
• Criteria for Similarity of Triangles

Basic Proportionality Theorem: Thales theorem is one of the most fundamental theorems in geometry that relates the parts of the length of sides of triangles. The other name of the Thales theorem is the Basic Proportionality Theorem or BPT.

BPT states that if a line is parallel to a side of a triangle that intersects the other sides into two distinct points, then the line divides those sides in proportion.

Let’s learn about the Thales Theorem or Basic Proportionality Theorem in detail, including its statement, proof, and converse as well.