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. https://i.imgur.com/1LU5Zmb.png
Source of the problem
Israel MO 2019 Problem 3
Topic
Geometry
Difficulty Level
6/10
Suggested Book
Challenges and Thrills of PreCollege Mathematics

Start with hints

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

Watch video

Connected Program at Cheenta

Math Olympiad Program

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

Squares and Triangles | AIME I, 2008 | Question 2

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Squares and Triangles.

Percentage Problem | AIME I, 2008 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Percentage. you may use sequential hints.

Smallest Positive Integer | PRMO 2019 | Question 14

Try this beautiful problem from the Pre-RMO, 2019 based on Smallest Positive Integer. You may use sequential hints to solve the problem.

Circles and Triangles | AIME I, 2012 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Circles and triangles.

Complex Numbers and Triangles | AIME I, 2012 | Question 14

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and Triangles.

Triangles and Internal bisectors | PRMO 2019 | Question 10

Try this beautiful problem from the Pre-RMO, 2019 based on Triangles and Internal bisectors. You may use sequential hints to solve the problem.

Angles in a circle | PRMO-2018 | Problem 80

Try this beautiful problem from PRMO, 2018 based on Angles in a circle. You may use sequential hints to solve the problem.

Digit Problem from SMO, 2012 | Problem 14

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2012 based on digit. You may use sequential hints to solve the problem.

Problem on Semicircle | AMC 8, 2013 | Problem 20

Try this beautiful problem from AMC-8, 2013, (Problem-20) based on area of semi circle.You may use sequential hints to solve the problem.

Radius of semicircle | AMC-8, 2013 | Problem 23

Try this beautiful problem from Geometry: Radius of semicircle from AMC-8, 2013, Problem-23. You may use sequential hints to solve the problem.