Cheenta is joining hands with Aditya Birla Education Academy for AMC Training.
Learn More

April 13, 2020

Centroid of Triangle | SMO, 2009 | Problem 1

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Centroid of Triangle.

Problem - Centroid of Triangle (SMO Entrance)

Let M and N be points on sides AB and AC of triangle ABC respectively. If \(\frac {BM}{MA} + \frac {CN}{NA} = 1\) . Can we show that MN passes through the centroid of ABC?

Key Concepts

2D - Geometry


Menelaus's Theorem

Check the Answer

Answer: Yes , we can.

Singapore Mathematics Olympiad

Challenges and Thrills - Pre College Mathematics

Try with Hints

If we got stuck in this problem then we can start this problem by applying Menelaus's Theorem.

Centroid of Triangle- Problem

It states : if a line intersects \(\triangle ABC\) or extended sides at points D, E, and F, the following statement holds: \(\frac {AD}{BD} \times \frac {BE}{EC} \times \frac {CF}{AF} = 1\)

Centroid of Triangle

Again let D is the mid point of AC. As \(\frac {BM}{MA} + \frac {CN}{NA} = 1\) then \(\frac {CN}{NA}<1\) where N lies in the line segment CD.From the picture above we can see g is the intersection point between two lines BD and MN. So if we apply Menelaus's Theorem we get :

\(\frac {DG}{GB} . \frac {BM}{MA} .\frac {AN}{ND} = 1\)

Now try the rest of the problem.....................................

After the 1st hint again, \(\frac {BG}{GD} = \frac{BM}{MA} . \frac {AN}{ND} = ( 1 - \frac {CN}{NA}). \frac {AN}{ND}\)

= \(\frac {NA - CN}{ND} = \frac {(2CD - CN) - CN}{ND}\)

=\(\frac {2 ND}{ND}\) = 2

Thus G is the centroid .(Proved)

Subscribe to Cheenta at Youtube

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.