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October 20, 2019

The geometry of circles from RMO 2019 Problem 5 Solution

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Understand the problem

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In an acute angled triangle ABC, let H be the orthocenter, and let D, E, F be the feet of the altitudes from A, B, C to the opposite sides, respectively. Let L, M, N be the midpoints of the segments AH, EF, BC respectively. Let X, Y be feet of altitudes from L, N on to the line DF. Prove that XM is perpendicular to MY

[/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="4.0" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" _builder_version="4.0" open="on"]Regional Math Olympiad, 2019 Problem 5[/et_pb_accordion_item][et_pb_accordion_item title="Topic" open="off" _builder_version="4.0"]Geometry

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[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.29.2" open="off"]Challenges and Thrills in Pre College 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"]

Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0"]Draw a diagram.  RMO 2019 problem 5 solution Construction: Can you find a circle? Nine-point circle passes through feet of altitudes, the midpoint of sides and midpoints of AH, BH, CH (where H is orthocenter) of a triangle. This is a well-known theorem from geometry.  In this case, it passes through L, E, D, N, and F.[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0"]

RMO 2019 Problem 5 circle Chase three cyclic quadrilaterals to understand DH is the internal angle bisector of \( \angle  EDF \). Since \( \angle BFE = \angle BEC = 90 ^o \) hence BFEC is cyclic hence \(\angle FBE = \angle FCE \) (angle subtended by the segment FE). Since \( \angle ADC = \angle AFC = 90 ^o \) hence AFDC is cyclic hence \(\angle FDA = \angle FCA or \angle FCE \) (angle subtended by the segment FA). Since \( \angle BDA = \angle BEA = 90 ^o \) hence BDEA is cyclic hence \(\angle ADE = \angle ABE \) (angle subtended by the segment AE). Combining we have \( \angle FDA = \angle EDA \) implying DA bisects \( \angle FDE \) rmo 2019 problem 5 angle bisector Can you show L-M-N are collinear and perpendicular to EF?

 

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0"]RMO 2019 Problem 5 straight line We show that \( \Delta LMF \) is congruent to \( \Delta LME \).  Clearly LM = LM, ME =  MF (as M is the midpoint). We will show that LE = LF. Why? Since L is the midpoint of LH and AFH is right-angled, hence L being the midpoint of hypotenuse is the circumcenter of AFH.  This implies LF = LA. Similarly, L is the circumcenter of AEH implying LE = LA.  Thus LF = LE.  Hence we showed that \( \Delta LMF \) is congruent to \( \Delta LME \) making \( \angle LMF = \angle LME = 90^o \) Similarly NM is perpendicular to EF.  Now can you find two cyclic quadrilaterals? [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0"]RMO 2019 Problem 5 straight line Notice that FXML is cyclic ( \( \angle FML = \angle FXL = 90^o \) ) Also, FMYN is cyclic ( \( \angle FMN = \angle FYN = 90^o \) ) We will show \( \angle FMX = \angle NMY \) (Then by adding \( \angle XMN \) to both side we will be done)  Toward that end we will show triangles FXM and NYM are similar.  \( \angle MFX  = \angle MNY \) (subtended by MY in cyclic quadrilateral MFNY ) \( \angle FXM = \angle NYM \) due to the following reason (making the triangles similar and remaining angle equal).  \( \angle FXM = \angle FXL + \angle LXM = 90^o + \angle LFM = 90^o + \angle LDE (green angle) \) \( \angle NYM = \angle NYF + \angle FYM = 90^o + \angle FNL = 90^o + \angle FDL (green angle) \) Hence proved.[/et_pb_tab][/et_pb_tabs][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"]

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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.[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][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="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems

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