  How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.0" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|8px|20px||" box_shadow_style="preset2"]Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line. [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.22.4" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.0"]Israel MO 2019 Problem 3[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.0" open="off"]Geometry[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.0" open="off"]6/10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0" open="off"]Challenges and Thrills of PreCollege Mathematics[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.22.4" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0"]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.[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0"]Let's investigate the triangle FDC. Observe that EF = AD and AD = AC. This results in the fact that FDC is isosceles and $\angle FDC = \pi - \angle EDA$. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0"]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. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0"]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

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