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Difference between revisions of "1995 USAMO Problems/Problem 3"

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Given a nonisosceles, nonright triangle <math>ABC,</math> let <math>O</math> denote the center of its circumscribed circle, and let <math>A_1, \, B_1,</math> and <math>C_1</math> be the midpoints of sides <math>BC, \, CA,</math> and <math>AB,</math> respectively.  Point <math>A_2</math> is located on the ray <math>OA_1</math> so that <math>\triangle OAA_1</math> is similar to <math>\triangle OA_2A</math>.  Points <math>B_2</math> and <math>C_2</math> on rays <math>OB_1</math> and <math>OC_1,</math> respectively, are defined similarly.  Prove that lines <math>AA_2, \, BB_2,</math> and <math>CC_2</math> are concurrent, i.e. these three lines intersect at a point.
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==Problem==
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Given a nonisosceles, nonright triangle <math>ABC,</math> let <math>O</math> denote its circumcenter, and let <math>A_1, \, B_1,</math> and <math>C_1</math> be the midpoints of sides <math>\overline{BC}</math>, <math>\overline{CA}</math>and <math>\overline{AB}</math> respectively.  Point <math>A_2</math> is located on the ray <math>OA_1</math> so that <math>\triangle OAA_1</math> is similar to <math>\triangle OA_2A</math>.  Points <math>B_2</math> and <math>C_2</math> on rays <math>OB_1</math> and <math>OC_1,</math> respectively, are defined similarly.  Prove that lines <math>AA_2, \, BB_2,</math> and <math>CC_2</math> are concurrent.
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== Solution ==
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'''LEMMA 1: ''' In <math>\triangle ABC</math> with circumcenter <math>O</math>, <math>\angle OAC = 90 - \angle B</math>.  
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'''PROOF of Lemma 1:''' The arc <math>AC</math> equals <math>2\angle B</math> which equals <math>\angle AOC</math>. Since <math>\triangle AOC</math> is isosceles we have that <math>\angle OAC = \angle OCA = 90 - \angle B</math>.
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QED
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--------------------------------------------------------------
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Define <math>H \in BC</math> s.t. <math>AH \perp BC</math>.  Since <math>OA_1 \perp BC</math>, <math>AH \parallel OA_1</math>.  Let <math>\angle AA_2O = \angle A_1AO = x</math> and <math>\angle AA_1O = \angle A_2AO = y</math>.  Since we have <math>AH \parallel OA_1</math>, we have that <math>\angle HAA_2 = x</math>.  Also, we have that <math>\angle A_2AA_1 = y-x</math>.  Furthermore, <math>\angle BAH = 90 - \angle B = \angle OAC</math>, by lemma 1.  Therefore, <math>\angle A_1AC = 90 - \angle B + x = \angle BAA_2</math>.  Since <math>A_1</math> is the midpoint of <math>BC</math>, <math>AA_1</math> is the median.  However <math>\angle A_1AC =  \angle BAA_2</math> tells us that <math>AA_2</math> is just <math>AA_1</math> reflected across the internal angle bisector of <math>A</math>.  By definition, <math>AA_2</math> is the <math>A</math>-symmedian.  Likewise, <math>BB_2</math> is the <math>B</math>-symmedian and <math>CC_2</math> is the <math>C</math>-symmedian.  Since the symmedians concur at the symmedian point, we are done.
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QED
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*[[Isogonal conjugate]]
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==See Also==
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{{USAMO box|year=1995|num-b=2|num-a=4}}
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{{MAA Notice}}
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[[Category:Olympiad Geometry Problems]]

Latest revision as of 00:42, 19 April 2024

Problem

Given a nonisosceles, nonright triangle $ABC,$ let $O$ denote its circumcenter, and let $A_1, \, B_1,$ and $C_1$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$and $\overline{AB}$ respectively. Point $A_2$ is located on the ray $OA_1$ so that $\triangle OAA_1$ is similar to $\triangle OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1,$ respectively, are defined similarly. Prove that lines $AA_2, \, BB_2,$ and $CC_2$ are concurrent.

Solution

LEMMA 1: In $\triangle ABC$ with circumcenter $O$, $\angle OAC = 90 - \angle B$.

PROOF of Lemma 1: The arc $AC$ equals $2\angle B$ which equals $\angle AOC$. Since $\triangle AOC$ is isosceles we have that $\angle OAC = \angle OCA = 90 - \angle B$. QED

Define $H \in BC$ s.t. $AH \perp BC$. Since $OA_1 \perp BC$, $AH \parallel OA_1$. Let $\angle AA_2O = \angle A_1AO = x$ and $\angle AA_1O = \angle A_2AO = y$. Since we have $AH \parallel OA_1$, we have that $\angle HAA_2 = x$. Also, we have that $\angle A_2AA_1 = y-x$. Furthermore, $\angle BAH = 90 - \angle B = \angle OAC$, by lemma 1. Therefore, $\angle A_1AC = 90 - \angle B + x = \angle BAA_2$. Since $A_1$ is the midpoint of $BC$, $AA_1$ is the median. However $\angle A_1AC = \angle BAA_2$ tells us that $AA_2$ is just $AA_1$ reflected across the internal angle bisector of $A$. By definition, $AA_2$ is the $A$-symmedian. Likewise, $BB_2$ is the $B$-symmedian and $CC_2$ is the $C$-symmedian. Since the symmedians concur at the symmedian point, we are done.

QED

See Also

1995 USAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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