Difference between revisions of "1959 IMO Problems/Problem 5"
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Quadrados e Circulos circunscritos / IMO 1959-#5 | Quadrados e Circulos circunscritos / IMO 1959-#5 | ||
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Link do vídeo: https://youtu.be/UNcHD5JI6wU | Link do vídeo: https://youtu.be/UNcHD5JI6wU | ||
− | {{IMO box|year=1959|num-b= | + | {{IMO box|year=1959|num-b=4|num-a=6}} |
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] | ||
[[Category:Geometric Construction Problems]] | [[Category:Geometric Construction Problems]] |
Revision as of 00:40, 2 January 2023
Problem
An arbitrary point is selected in the interior of the segment . The squares and are constructed on the same side of , with the segments and as their respective bases. The circles about these squares, with respective centers and , intersect at and also at another point . Let denote the point of intersection of the straight lines and .
(a) Prove that the points and coincide.
(b) Prove that the straight lines pass through a fixed point independent of the choice of .
(c) Find the locus of the midpoints of the segments as varies between and .
Solution
Part A
Since the triangles are congruent, the angles are congruent; hence is a right angle. Therefore must lie on the circumcircles of both quadrilaterals; hence it is the same point as .
Part B
We observe that since the triangles are similar. Then bisects .
We now consider the circle with diameter . Since is a right angle, lies on the circle, and since bisects , the arcs it intercepts are congruent, i.e., it passes through the bisector of arc (going counterclockwise), which is a constant point.
Part C
Denote the midpoint of as . It is clear that 's distance from is the average of the distances of and from , i.e., half the length of , which is a constant. Therefore the locus in question is a line segment.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
Quadrados e Circulos circunscritos / IMO 1959-#5 Link do vídeo: https://youtu.be/UNcHD5JI6wU
1959 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |