2005 Indonesia MO Problems/Problem 7
Problem
Let be a convex quadrilateral. Square is constructed such that the two vertices is located outside . Similarly, we construct squares , , . Let be the intersection of and , be the intersection of and , be the intersection of and , and be the intersection of and . Prove that is perpendicular to .
Solution
Let the coordinates of be , the coordinates of be , the coordinates of be , and the coordinates of be , where all variables are rational and .
Let be the midpoint of , which is point . Additionally, mark points , , and .
Note that since is the center of square , and . Additionally, and .
is a line, so . Since , , so . Additionally, because is a right triangle, . Rearranging and substituting results in .
Since both and are right triangles, by AAS Congruency, . Therefore and . From this information, the coordinates of are .
By using similar reasoning, the coordinates of are , the coordinates of are , and the coordinates of are .
The slope of is . The slope of is . The product of the two slopes is , so .
See Also
2005 Indonesia MO (Problems) | ||
Preceded by Problem 6 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 8 |
All Indonesia MO Problems and Solutions |