Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

 
(→‎Solution)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
−
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.
+
The vertices of a right triangle <math>ABC</math> inscribed in a circle divide the circumference into three arcs.
 +
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
 +
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,
 +
the point <math>D</math> on arc <math>BC</math> is the midpoint of the segment joining the points <math>D^\prime</math>
 +
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>.
 +
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.
  
−
{{solution}}
+
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:
 +
<cmath>
 +
\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*}
 +
</cmath>
 +
whence we conclude that <math>\angle{EDF} = 60^\circ.</math>
 +
 +
Next, we prove that triangle <math>DYF</math> is equilateral. To see this, note that
 +
<cmath>
 +
\begin{align*}
 +
\angle{DYF} &= \angle{FAB} + \angle{BAD} \\
 +
                    &= \angle{FDY} \\
 +
                    &= \angle{YFD}.
 +
\end{align*}
 +
</cmath>
 +
Hence <math>\angle{FED} = 60^\circ</math> as well, so triangle <math>DEF</math> is equilateral as desired.
 +
 +
<math>\blacksquare</math>
 +
 +
==See also==
 
*[[2006 Canadian MO]]
 
*[[2006 Canadian MO]]
 +
 +
{{CanadaMO box|year=2006|num-b=4|after=Last question}}

Latest revision as of 16:02, 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
Last question
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_Canadian_MO_Problems/Problem_5&oldid=39070"