Difference between revisions of "2023 IMO Problems/Problem 6"
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==Problem== | ==Problem== | ||
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Let <math>ABC</math> be an equilateral triangle. Let <math>A_1,B_1,C_1</math> be interior points of <math>ABC</math> such that <math>BA_1=A_1C</math>, <math>CB_1=B_1A</math>, <math>AC_1=C_1B</math>, and | Let <math>ABC</math> be an equilateral triangle. Let <math>A_1,B_1,C_1</math> be interior points of <math>ABC</math> such that <math>BA_1=A_1C</math>, <math>CB_1=B_1A</math>, <math>AC_1=C_1B</math>, and | ||
<cmath>\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ</cmath>Let <math>BC_1</math> and <math>CB_1</math> meet at <math>A_2,</math> let <math>CA_1</math> and <math>AC_1</math> meet at <math>B_2,</math> and let <math>AB_1</math> and <math>BA_1</math> meet at <math>C_2.</math> | <cmath>\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ</cmath>Let <math>BC_1</math> and <math>CB_1</math> meet at <math>A_2,</math> let <math>CA_1</math> and <math>AC_1</math> meet at <math>B_2,</math> and let <math>AB_1</math> and <math>BA_1</math> meet at <math>C_2.</math> | ||
Revision as of 03:42, 24 July 2023
Problem
Let 
be an equilateral triangle. Let 
be interior points of such that 
, 
, 
, and
![\[\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ\]](http://latex.artofproblemsolving.com/1/6/1/16178c056d256d3702ee4b501044e9de5f4e9e5b.png)
Let 
and 
meet at 
let 
and 
meet at 
and let 
and 
meet at 
Prove that if triangle 
is scalene, then the three circumcircles of triangles 
and 
all pass through two common points.
(Note: a scalene triangle is one where no two sides have equal length.)
Solution
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]
