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Difference between revisions of "2023 IMO Problems/Problem 6"

(Created page with "==Problem== 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...")
 
Line 2: Line 2:
 
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>
 +
 
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 06:14, 14 July 2023

Problem

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and \[\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ\]Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$

Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ 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]

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