Difference between revisions of "2023 IMO Problems/Problem 6"
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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> | ||
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Prove that if triangle <math>A_1B_1C_1</math> is scalene, then the three circumcircles of triangles <math>AA_1A_2, BB_1B_2</math> and <math>CC_1C_2</math> all pass through two common points. | Prove that if triangle <math>A_1B_1C_1</math> is scalene, then the three circumcircles of triangles <math>AA_1A_2, BB_1B_2</math> and <math>CC_1C_2</math> all pass through two common points. | ||
Revision as of 07:14, 14 July 2023
Problem
Let be an equilateral triangle. Let
be interior points of
such that
,
,
, and
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]