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Difference between revisions of "2006 Canadian MO Problems/Problem 5"

 
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==Problem==
 
==Problem==
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The vertices of right triangle <math>ABC</math> inscribed in a circle divide the three arcs, we draw a tangent intercepted by the lines <math>AB</math> and <math>AC</math>. If the tangency points are <math>D</math>, <math>E</math>, and <math>F</math>, show that the triangle <math>DEF</math> is equilateral.
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The vertices of a right triangle <math>ABC</math> inscribed in a circle divide the circumference into three arcs.
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The right angle is at <math>A</math>, so that the opposite arc <math>BC</math> is a semicircle while arc <math>AB</math> and arc <math>AC</math> are
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supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the
 +
midpoint of that portion of the tangent intercepted by the extended lines <math>AB</math> and <math>AC</math>. More precisely,
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the point <math>D</math> on arc <math>BC</math> is the midpoint of the segment joining the points <math>D^\prime</math>
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and <math>D^\prime^\prime</math> where the tangent at <math>D</math> intersects the extended lines <math>AB</math> and <math>AC</math>. Similarly for <math>E</math> on arc <math>AC</math> and <math>F</math> on arc <math>AB</math>.
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Prove that triangle <math>DEF</math> is equilateral.
 +
 
 
==Solution==
 
==Solution==
 +
Let the intersection of the tangents at <math>D</math> and <math>E</math>, <math>E</math> and <math>F</math>, <math>F</math> and <math>D</math> be labeled <math>Z, X,Y</math>, respectively.
  
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{{solution}}
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It is a well-known fact that in a right triangle <math>PQR</math> with <math>M</math> the midpoint of hypotenuse <math>PR</math>, triangles <math>MQR</math> and <math>PQM</math> are isosceles.
  
 +
Now we do some angle-chasing:
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<cmath>
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\begin{align*}
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\angle{EDF} &= \angle{EDA} + \angle{ADF} \
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                    &= \angle{XEA} + \angle{AFX} \
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                    &= (180^\circ - \angle{AEZ}) + (180^\circ - \angle{YFA}) \
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                    &= 2\angle{FAB} + 2\angle{CAE}\
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                    &= 2(\angle{FAE} - 90^\circ)\
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                    &= 2(90^\circ - \angle{EDF}),
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\end{align*}
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</cmath>
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whence we conclude that <math>\angle{EDF} = 60^\circ.</math>
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Next, we prove that triangle <math>DYF</math> is equilateral. To see this, note that
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<cmath>
 +
\begin{align*}
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\angle{DYF} &= \angle{FAB} + \angle{BAD} \
 +
                    &= \angle{FDY} \
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                    &= \angle{YFD}.
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\end{align*}
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</cmath>
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Hence <math>\angle{FED} = 60^\circ</math> as well, so triangle <math>DEF</math> is equilateral as desired.
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<math>\blacksquare</math>
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 +
==See also==
 
*[[2006 Canadian MO]]
 
*[[2006 Canadian MO]]
 +
 +
{{CanadaMO box|year=2006|num-b=4|after=Last question}}

Latest revision as of 17:02, 3 June 2011

Problem

The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite arc $BC$ is a semicircle while arc $AB$ and arc $AC$ are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines $AB$ and $AC$. More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D^\prime$ and $D^\prime^\prime$ (Error compiling LaTeX. Unknown error_msg) where the tangent at $D$ intersects the extended lines $AB$ and $AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$. Prove that triangle $DEF$ is equilateral.

Solution

Let the intersection of the tangents at $D$ and $E$, $E$ and $F$, $F$ and $D$ be labeled $Z, X,Y$, respectively.

It is a well-known fact that in a right triangle $PQR$ with $M$ the midpoint of hypotenuse $PR$, triangles $MQR$ and $PQM$ are isosceles.

Now we do some angle-chasing: \begin{align*} \angle{EDF} &= \angle{EDA} + \angle{ADF} \\ &= \angle{XEA} + \angle{AFX} \\ &= (180^\circ - \angle{AEZ}) + (180^\circ - \angle{YFA}) \\ &= 2\angle{FAB} + 2\angle{CAE}\\ &= 2(\angle{FAE} - 90^\circ)\\ &= 2(90^\circ - \angle{EDF}), \end{align*} whence we conclude that $\angle{EDF} = 60^\circ.$

Next, we prove that triangle $DYF$ is equilateral. To see this, note that \begin{align*} \angle{DYF} &= \angle{FAB} + \angle{BAD} \\ &= \angle{FDY} \\ &= \angle{YFD}. \end{align*} Hence $\angle{FED} = 60^\circ$ as well, so triangle $DEF$ is equilateral as desired.

$\blacksquare$

See also

2006 Canadian MO (Problems)
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 Followed by
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