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 |