2006 Canadian MO Problems/Problem 2
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 .
We claim that the desired locus is the line segment from the midpoint of altitude to the midpoint of , , not including both endpoints.
A homothety about maps the rectangle onto rectangle in the exterior of . The scale factor of the homothety is , which is also the scale factor of the mapping of the intersection of diagonals (the original we call and the new we call . Hence . But , and , so and are similar, and so lies on , as desired. Reversing the argument proves the other direction for a locus, and we are done.
|2006 Canadian MO (Problems)|
|1 • 2 • 3 • 4 • 5||Followed by|