Try this beautiful problem. We will give you hints!
ABC is an equilateral triangle with vertex A fixed and B moving in a given straight line. Find the locus of C
[/et_pb_text][et_pb_accordion _builder_version="3.22.1" body_font="Raleway||||||||" toggle_font="||||||||"][et_pb_accordion_item title="What is the big idea ?" open="on" _builder_version="3.22.1" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" body_text_shadow_horizontal_length="0em" body_text_shadow_vertical_length="0em" body_text_shadow_blur_strength="0em"]
Think about what does not change.
1. Point A is fixed.
2. The line on which B moves is fixed.
3. The fact that ABC is always equilateral is fixed.
1. The position of C will change (as B moves).
2. Length of the sides of the triangle will change (it needs to stay equilateral).
Carefully draw several positions of B (and corresponding positions of C). This will give you an idea about what is the possible path traced out by C
[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.22.1" text_font="Raleway||||||||" border_radii="on|5px|5px|5px|5px" border_width_all="1px" border_color_all="#0c71c3" box_shadow_style="preset4" text_orientation="center" custom_padding="20px||20px"]This is problem is a part of the Math Olympiad Program and I.S.I. Entrance Program at Cheenta.
[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" button_text="Learn More" _builder_version="3.22.1" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_font="||||||||" box_shadow_style="preset1" transform_translate="37px|-38px" background_layout="dark"][/et_pb_button][/et_pb_column][et_pb_column type="1_2" _builder_version="3.22.1"][et_pb_text _builder_version="3.22.1" text_font="Raleway||||||||"]
1. Challenges and Thrills of Pre College Mathematics.
2. Geometry Revisited by Coxeter.
Drop a perpendicular from A on the line L. Suppose the feet of the perpendicular is P.
Make AB such that \( \angle BAP = 30^o \)
Reflect BAP about AP. Let the image of B under reflection be C.
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.22.1"]Draw a line through C that makes an angle of \( 60^0 \) with L. We will call this Red line.
Claim: This Red Line is part of the locus of point C.
[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.22.1"]Pick another point \( B_1 \) on L.
Joint \( AB_1 \)
Make angle \( 60^0 \) with \( AB_1 \) and draw a line. Let it hit the red line at \( C_1 \)
Show that \( AB_1 C_1 \) is equilateral.
[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.22.1"]Notice that \( AB_1 C C_1 \) is cyclic. Why? After all \( \angle B_1 A C_1 = 60^0 \) and \( \angle B_1 C C_1 = 120^o \). Hence opposite angles add up to \( 180^o \).
Any quadrilateral in which opposite angles add up to \( 180^o \) is cyclic.
This immediately proves \( AB_1 C_1 \) is equilateral. Hence red line is part of the locus.
Find the other part of the locus by interchanging the positions of B and C n first step.
[/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
locus of c is the straight line through c making 60 degrees angle with the base BC produced. easy one