Menelaus Theorem

Menelaus Theorem is named for Menelaus of Alexandria, a Greek mathematician, and astronomer. The internal and external division concepts of the triangle with proof are given below:

Menelaus Theorem
Menelaus Theorem

In the given figure, a transversal line (Red) is drawn, cutting sides of the ∆ABC at D, E & F.

Menelaus Theorem Internal division
Internal division

Construct perpendiculars AA’, BB’, and CC’. 

Now, Since ∆AA’D ~ ∆BB’D (Similar) 

BD/DA=BB’/AA’

Since ∆BB’E ~ ∆CC’E

CE/EB=CC’/BB’

Since ∆AA’F ~ ∆CC’F

AF/CF=AA’/CC’

(AF/CF)×(CE/EB)×(BD/DA)=(AA’/CC’)×(CC’/BB’)×(BB’/AA’)

(AF/CF)×(CE/EB)×(BD/DA)=1   …………….(i)


Equality still holds even when the Red Line does not intersect the triangle at all. 

Menelaus Theorem external division
External division

Construct perpendiculars AA’, BB’, and CC’. 

Menelaus Theorem proof
Theorem’s Proof

Now, Since ∆AA’F ~ ∆CC’F (Similar) 

AF/CF=AA’/CC’

Since ∆BB’D ~ ∆AA’D

BD/AD=BB’/AA’

Since ∆BB’E ~ ∆CC’E

CE/EB=CC’/BB’

(AF/CF)×(CE/EB)×(BD/DA)=(AA’/CC’)×(CC’/BB’)×(BB’/AA’)

(AF/CF)×(CE/EB)×(BD/DA)=1   …………….(ii)

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Application of Menelaus Theorem

Viral Geometry problem (Menelaus Theorem)
Viral Geometry problem

In the given figure, if PT: TU=2 : 3 and PU : UR=4: 5 then find TS: SR and US: SQ?

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Step :1

Menelaus Theorem:1
Step-1

(PQ/TQ)×(TS/SR)×(RU/UP)=1

(5/3)×(TS/SR)×(5/4)=1

TS: SR=12: 25 ……………..(i)

Step:2 

Menelaus Theorem:2
Step-2

(PR/UR)×(US/SQ)×(QT/TP)=1 

(9/5)×(US/SQ)×(3/2)=1 

US : SQ=10 : 27  ……………..(ii)

Inradius and Circumradius of the Triangle: Click Here

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