be a triangle, 
its circumcenter, 
its centroid, and 
its orthocenter. Denote by 
, and 
the centers of the circles circumscribed about the triangles 
, and 
, respectively. Prove that the triangle is congruent to the triangle 
and that the nine-point circle of 
is also the nine-point circle of 
.[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.22.4"][et_pb_column type="4_4" _builder_version="3.22.4"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.23.3" 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="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]IMO longlist 1992[/et_pb_accordion_item][et_pb_accordion_item title="Topic" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Geometry[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Hard[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Challenge and Thrill of Pre-college Mathematics[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]From the last hint, we have $latex BA_1=BO=R=CO=CA_1$ where $latex R$ is the circumradius of $latex ABC$. Hence, $latex BOCA_1$ is a rhombus and $latex A_1C||BO$. Similarly, $latex BO||AC_1$ hence $latex A_1C||AC_1$. As $latex CAC_1A_1$ is a parallelogram, we also have $latex A_1C_1=AC$. We can similarly prove that $latex A_1B_1=AB$ and $latex B_1C_1=BC$. Thus $latex ABC\cong A_1B_1C_1$. This implies that it suffices to show that the centres of the two nine-point circles coincide. Remember that the nine-point centre is the midpoint of the line joining the circumcentre and the orthocentre. Claim $latex H$ is the circumcentre of $latex A_1B_1C_1$. Proof Note that $latex A_1H=R$ (as $latex A_1$ is the centre of $latex \odot BHC$ and $latex \odot BHC$ is a reflection of $latex \odot ABC$. Similarly, $latex B_1H=C_1H=R$. Claim $latex O$ is the orthocentre of $latex A_1B_1C_1$. Proof As $latex A_1$ is the reflection of $latex O$ on $latex BC$, $latex A_1O\perp BC$. As $latex BC\parallel B_1C_1$, $latex A_1O\perp B_1C_1$. Similarly, $latex B_1O\perp C_1A_1$ and $latex C_1O\perp A_1B_1$. Thus $latex O$ is the orthocentre of $latex A_1B_1C_1$.
Thus the centres of the nine-point circles of $latex ABC$ and $latex A_1B_1C_1$ coincide.
[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]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://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%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark"][/et_pb_button][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px"]
