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
Difficulty Level
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 \( \angle EFM = \angle DFC\).


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