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 of the triangle (Powerful Geometry hack 2022) 2 Menelaus Theorem](https://logicxonomy.com/wp-content/uploads/2022/09/1-1024x576.webp)
In the given figure, a transversal line (Red) is drawn, cutting sides of the ∆ABC at D, E & F.
![Menelaus Theorem of the triangle (Powerful Geometry hack 2022) 3 Menelaus Theorem Internal division](https://logicxonomy.com/wp-content/uploads/2022/09/2-1024x576.webp)
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 of the triangle (Powerful Geometry hack 2022) 4 Menelaus Theorem external division](https://logicxonomy.com/wp-content/uploads/2022/09/3-1024x576.webp)
Construct perpendiculars AA’, BB’, and CC’.
![Menelaus Theorem of the triangle (Powerful Geometry hack 2022) 5 Menelaus Theorem proof](https://logicxonomy.com/wp-content/uploads/2022/09/4-1024x576.webp)
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)
![Menelaus Theorem of the triangle (Powerful Geometry hack 2022) 6 Online Class Logicxonomy](https://logicxonomy.com/wp-content/uploads/2023/03/20230314_070946_0000-1024x1024.webp)
Application of Menelaus Theorem
![Menelaus Theorem of the triangle (Powerful Geometry hack 2022) 7 Viral Geometry problem (Menelaus Theorem)](https://logicxonomy.com/wp-content/uploads/2022/09/5-1024x576.webp)
In the given figure, if PT: TU=2 : 3 and PU : UR=4: 5 then find TS: SR and US: SQ?
Step :1
![Menelaus Theorem of the triangle (Powerful Geometry hack 2022) 9 Menelaus Theorem:1](https://logicxonomy.com/wp-content/uploads/2022/09/6-1024x576.webp)
(PQ/TQ)×(TS/SR)×(RU/UP)=1
(5/3)×(TS/SR)×(5/4)=1
TS: SR=12: 25 ……………..(i)
Step:2
![Menelaus Theorem of the triangle (Powerful Geometry hack 2022) 10 Menelaus Theorem:2](https://logicxonomy.com/wp-content/uploads/2022/09/7-1024x576.webp)
(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