AMC-8 USA Math Olympiad

Menalaus Theorem in AMC 8 2019

What are we learning ?

Competency in Focus: Menalaus’s Theorem This problem from American Mathematics contest (AMC 8, 2019) will help us to learn more about Menalaus’s Theorem. 

First look at the knowledge graph.

Next understand the problem

In triangle 𝐴𝐵𝐶, point 𝐷 divides side AC so that 𝐴𝐷 ∶ 𝐷𝐶 = 1 ∶ 2. Let 𝐸 be the midpoint of BD and 𝐹 be the point of intersection of line BC and line AE. Given that the area of ∆𝐴𝐵𝐶 is 360, what is the area of ∆𝐸𝐵𝐹?
Source of the problem
American Mathematical Contest 2019, AMC 8 Problem 25

Key Competency
Menalaus’s Theorem:   Given a triangle ABC, and a transversal line that crosses BC, AC, and AB at points D, E, and F respectively, with D, E, and F distinct from A, B, and C, then

$$ \displaystyle {\frac {AF}{FB}\times \frac {BD}{DC}\times \frac {CE}{EA}=-1.}$$

Difficulty Level


Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

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