Difference between revisions of "1959 IMO Problems/Problem 5"
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Denote the midpoint of <math>PQ </math> as <math>R </math>. It is clear that <math>R </math>'s distance from <math>AB </math> is the average of the distances of <math>P </math> and <math>Q </math> from <math>AB </math>, i.e., half the length of <math>AB</math>, which is a constant. Therefore the locus in question is a line segment. | Denote the midpoint of <math>PQ </math> as <math>R </math>. It is clear that <math>R </math>'s distance from <math>AB </math> is the average of the distances of <math>P </math> and <math>Q </math> from <math>AB </math>, i.e., half the length of <math>AB</math>, which is a constant. Therefore the locus in question is a line segment. | ||
+ | == Solution 2 == | ||
+ | [[Image:IMO19595A.png]] | ||
+ | === Part a) === | ||
+ | Notice that <math>\angle BAF = \arctan (\frac{MF}{AM}) = \frac{MB}{AM} </math> and <math>\angle ABC = \arctan (\frac{MC}{MB}) = \frac{AM}{MB} </math>. | ||
− | {{ | + | <math>\implies </math> <math>\angle BAF + \angle ABC = 90^{\circ} </math>. |
+ | |||
+ | Now notice that <math>\angle FNB = 90^{\circ} </math>. Considering <math>\angle BAF </math> as <math>\theta </math>, this gives <math>\angle MBN = 90^{\circ} - \theta </math> and thus <math>\angle MFN = 90^{\circ} + \theta </math>. But notice that <math>\angle MFA = 90^{\circ} - \theta </math>, which means that <math>\angle AFN = 180^{\circ} </math>. Therefore points <math>A, F, N </math> are collinear. Now <math>\angle BNF = 90^{\circ} </math> and <math>\angle ANC = 90^{\circ} </math>. Therefore, <math>\angle BNC = 180^{\circ} </math> and thus points <math>B, N, C </math> are collinear. Therefore, AF and BC intersect at N. | ||
+ | |||
+ | === Part b) === | ||
+ | Construct the bisector of arc AB above AB. Call it X. <math>\angle ANM = \angle ACM = 45^{\circ} </math>. Now <math>\angle ANB = 90^{\circ}</math> which means N lies on the circle with AB as diameter. | ||
+ | |||
+ | <math>\implies </math> <math>\angle ANX = \angle ABX = 45^{\circ} = \angle ANM </math>. Therefore since M and X are on the same side of <math>N </math>, <math>NM </math> passes through <math>X </math> wherever we choose <math>M </math> on <math>AB </math>. | ||
+ | |||
+ | === Part c) === | ||
+ | Let the midpoint of <math>PQ </math> be <math>R </math>. Let <math>G </math> be the midpoint of <math>AM </math>. Let <math>F </math> be the midpoint of <math>MB </math>. Let <math>I </math> be the foot of the perpendicular from <math>R </math> onto <math>AB </math>. Therefore by Midpoint Theorem, <math>RI = \frac{PG + HQ}{2} = \frac{AG + HB}{2} = \frac{AB}{4} </math>. Therefore the distance <math>RI </math> is a constant and thus the locus is a straight line parallel to <math>AB </math> at a distance (to <math>AB </math>, of course) of <math>\frac{AB}{4} </math>. | ||
== See Also == | == See Also == | ||
+ | Quadrados e Circulos circunscritos / IMO 1959-#5 | ||
+ | Link do vídeo: https://youtu.be/UNcHD5JI6wU | ||
{{IMO box|year=1959|num-b=4|num-a=6}} | {{IMO box|year=1959|num-b=4|num-a=6}} | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] | ||
+ | [[Category:Geometric Construction Problems]] |
Latest revision as of 07:23, 23 May 2024
Contents
[hide]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.
Solution 2
Part a)
Notice that and .
.
Now notice that . Considering as , this gives and thus . But notice that , which means that . Therefore points are collinear. Now and . Therefore, and thus points are collinear. Therefore, AF and BC intersect at N.
Part b)
Construct the bisector of arc AB above AB. Call it X. . Now which means N lies on the circle with AB as diameter.
. Therefore since M and X are on the same side of , passes through wherever we choose on .
Part c)
Let the midpoint of be . Let be the midpoint of . Let be the midpoint of . Let be the foot of the perpendicular from onto . Therefore by Midpoint Theorem, . Therefore the distance is a constant and thus the locus is a straight line parallel to at a distance (to , of course) of .
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 |