Difference between revisions of "2024 USAMO Problems/Problem 5"
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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 of triangle <math>BEM</math>. | + | 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 05:27, 1 November 2024
- The following problem is from both the 2024 USAMO/5 and 2024 USAJMO/6, so both problems redirect to this page.
Contents
Problem
Point is selected inside acute triangle
so that
and
. Point
is chosen on ray
so that
. Let
be the midpoint of
. Show that line
is tangent to the circumcircle of triangle
..
Solution 1
Let and
.
Extend AD intersects BC at point T, then TC = TA, TE is perpendicular to AC
Thus, AB is the tangent of the circle BEM
Then the question is equivalent as the is the auxillary angle of
.
continue
See Also
2024 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.