2003 Indonesia MO Problems/Problem 2
Contents
[hide]Problem
Given a quadrilateral . Let
,
,
, and
are the midpoints of
,
,
, and
, respectively.
and
intersects at
. Prove that
and
.
Solution 1
Draw lines and
. By SAS Similarity,
and
That means
and
making
a parallelogram.
Since is a parallelogram,
In addition, by the Alternating Interior Angle Theorem,
and
Thus, by ASA Congruency,
Finally, using CPCTC shows that
and
Solution 2
Let ,
,
, and
. Then, we have
,
,
, and
. Note that any line that goes through two points
and
also goes through their midpoint. So, the line through
and
also goes through
. Similarly, any line through
and
also goes through
. That means that both
and
go through
, and because two non-identical lines that intersect only intersect once,
. Since
is the midpoint of both
and
, we have proved that
and
.
~Puck_0
See Also
2003 Indonesia MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 3 |
All Indonesia MO Problems and Solutions |