Select Page

# Understand the problem

Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line.
##### Source of the problem
Israel MO 2019 Problem 3
Geometry
6/10
##### Suggested Book
Challenges and Thrills of PreCollege Mathematics

Do you really need a hint? Try it first!

You need to show that M, F, C lie on a straight line. Observe that it can be shown that they are collinear if we can show that $\angle EFM = \angle CFD$.  We will now proceed towards proving in this direction.
Let’s investigate the triangle FDC. Observe that EF = AD and AD = AC. This results in the fact that FDC is isosceles and $\angle FDC = \pi – \angle EDA$.
Now, we will try to infer something about triangle MEF. Observe that KM || HI as $\angle MKJ = \angle KJH$. Hence KHIM must be a parallelogram. Hence, KH || MI. Also, $\angle KHI =\angle HIG$.  Hence, KH || EG.  Hence, it implies from  KH || MI and KH || EG, that M,E,I,G are collinear.
Now, this diagram ends it all. Observe that MI = KH. Also, EI = LJ. Hence, ME = KL = EP. Hence, MEF is isosceles.  Also, $\angle MEF = \pi – \angle GEF = \pi – \angle EDA = \angle FDC$. Hence, triangle MEF is similar to triangle FDC. This implies that $\angle EFM = \angle DFC$.

QED

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

## Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

## Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

## Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

## Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

## Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.

## Rectangular Piece of Paper | AMC 10A, 2014| Problem No 22

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2014. Problem-23. You may use sequential hints to solve the problem.

## Probability in Marbles | AMC 10A, 2010| Problem No 23

Try this beautiful Problem on Probability from AMC 10A, 2010. Problem-23. You may use sequential hints to solve the problem.

## Points on a circle | AMC 10A, 2010| Problem No 22

Try this beautiful Problem on Number theory based on Triangle and Circle from AMC 10A, 2010. Problem-22. You may use sequential hints to solve the problem.

## Circle and Equilateral Triangle | AMC 10A, 2017| Problem No 22

Try this beautiful Problem on Triangle and Circle from AMC 10A, 2017. Problem-22. You may use sequential hints to solve the problem.