2006 Canadian MO Problems/Problem 5
Problem
The vertices of a right triangle inscribed in a circle divide the circumference into three arcs.
The right angle is at
, so that the opposite arc
is a semicircle while arc
and arc
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
and
. More precisely,
the point
on arc
is the midpoint of the segment joining the points
and $D^\prime^\prime$ (Error compiling LaTeX. Unknown error_msg) where the tangent at
intersects the extended lines
and
. Similarly for
on arc
and
on arc
.
Prove that triangle
is equilateral.
Solution
Let the intersection of the tangents at and
,
and
,
and
be labeled
, respectively.
It is a well-known fact that in a right triangle
with
the midpoint of hypotenuse
, triangles
and
are isosceles.
Now, we do some angle-chasing:
whence we conclude that
Next, we will prove that triangle
is equilateral. To see this, note that
Then,
as well, and we are done.
See also
2006 Canadian MO (Problems) | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 | Followed by Last question |