2006 Canadian MO Problems/Problem 2
Problem
Let be an acute angled triangle. Inscribe a rectangle
in this triangle so that
is on
,
on
, and
and
on
. Describe the locus of the intersections of the diagonals of all possible rectangles
.
Solution
The locus is the line segment which joins the midpoint of side to the midpoint of the altitude to side
of the triangle.
Let and let
be the foot of the altitude from
to
. Then by similarity,
.
Now, we use vector geometry: intersection of the diagonals of
is also the midpoint of diagonal
, so
,
and this point lies on the segment joining the midpoint of segment
and the midpoint
of segment
, dividing this segment into the ratio
.
See also
2006 Canadian MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 | Followed by Problem 3 |