Difference between revisions of "2024 USAJMO Problems/Problem 6"

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== Problem ==
 
== Problem ==
Point <math>D</math> is selected inside acute triangle <math>ABC</math> so that <math>\angle DAC=\angle ACB</math> and <math>\angle BDC=90^\circ+\angle BAC</math>. Point <math>E</math> is chosen on ray <math>BD</math> so that <math>AE=EC</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Show that line <math>AB</math> is tangent to the circumcircle.
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Point <math>D</math> is selected inside acute triangle <math>ABC</math> so that <math>\angle DAC=\angle ACB</math> and <math>\angle BDC=90^\circ+\angle BAC</math>. Point <math>E</math> is chosen on ray <math>BD</math> so that <math>AE=EC</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Show that line <math>AB</math> is tangent to the circumcircle of triangle <math>BEM</math>.
  
 
== Solution 1 ==
 
== Solution 1 ==

Revision as of 12:42, 23 March 2024

Problem

Point $D$ is selected inside acute triangle $ABC$ so that $\angle DAC=\angle ACB$ and $\angle BDC=90^\circ+\angle BAC$. Point $E$ is chosen on ray $BD$ so that $AE=EC$. Let $M$ be the midpoint of $BC$. Show that line $AB$ is tangent to the circumcircle of triangle $BEM$.

Solution 1

See Also

2024 USAJMO (ProblemsResources)
Preceded by
Problem 5
Followed by
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of triangle $BEM$.