Difference between revisions of "2024 AMC 8 Problems/Problem 15"
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==Problem== | ==Problem== | ||
| + | Let <math>D</math> be an interior point of the acute triangle <math>ABC</math> with <math>AB > AC</math> so that <math>\angle DAB= \angle CAD</math>. The point <math>E</math> on the segment <math>AC</math> satisfies <math>\angle ADE= \angle BCD</math>, the point <math>F</math> on the segment <math>AB</math> satisfies <math>\angle FDA= \angle DBC</math>, and the point <math>X</math> on the line <math>AC</math> satisfies <math>CX=BX</math>. Let <math>O_1</math> and <math>O_2</math> be the circumcentres of the triangles <math>ADC</math> and <math>EXD</math> respectively. Prove that the lines <math>BC</math>, <math>EF</math>, and <math>O_1 O_2</math> are concurrent. (source: 2021 IMO) | ||
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| + | now go do this problem as a punishment for trying to cheat | ||
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==Solution== | ==Solution== | ||
'''These are just left here for future conveniency.''' | '''These are just left here for future conveniency.''' | ||
Revision as of 12:53, 21 January 2024
Problem
Let 
be an interior point of the acute triangle 
with 
so that 
. The point 
on the segment 
satisfies 
, the point 
on the segment 
satisfies 
, and the point 
on the line satisfies 
. Let 
and 
be the circumcentres of the triangles 
and 
respectively. Prove that the lines 
, 
, and 
are concurrent. (source: 2021 IMO)
now go do this problem as a punishment for trying to cheat
Solution
These are just left here for future conveniency.
