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"

m (→‎Solution: box)
(→‎Problem)
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==
 
{{solution}}
 
{{solution}}

Revision as of 19:03, 2 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

This problem needs a solution. If you have a solution for it, please help us out by adding it.

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=38974"