Difference between revisions of "1959 IMO Problems/Problem 5"
(→Part a)) |
(→Part a)) |
||
Line 30: | Line 30: | ||
[[Image:IMO19595A.png]] | [[Image:IMO19595A.png]] | ||
=== Part a) === | === Part a) === | ||
− | Notice that <math>\angle BAF = | + | Notice that <math>\angle BAF = \arctan (\frac{MF}{AM}) = \frac{MB}{AM} </math> and <math>\angle ABC = </math>arctan (\frac{MC}{MB}) = \frac{AM}{MB} <math>. |
− | <math>\implies < | + | </math>\implies <math> </math>\angle BAF + \angle ABC = 90^{\circ} <math>. |
− | Now notice that <math>\angle FNB = 90^{\circ} < | + | 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} and \angle ANC = 90^{\circ} <math>. Therefore, </math>\angle BNC = 180^{\circ} <math> and thus points </math>B, N, C $ are collinear. Therefore, AF and BC intersect at N. |
=== Part b) === | === Part b) === |
Revision as of 08:18, 23 May 2024
Contents
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
arctan (\frac{MC}{MB}) = \frac{AM}{MB}
\implies $$ (Error compiling LaTeX. Unknown error_msg)\angle BAF + \angle ABC = 90^{\circ} $.
Now notice that$ (Error compiling LaTeX. Unknown error_msg)\angle FNB = 90^{\circ} \angle BAF
\theta
\angle MBN = 90^{\circ} - \theta
\angle MFN = 90^{\circ} + \theta
\angle MFA = 90^{\circ} - \theta
\angle AFN = 180^{\circ}
A, F, N
\angle BNF = 90^{\circ} and \angle ANC = 90^{\circ}
\angle BNC = 180^{\circ}
B, N, C $ 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
.
%alternate solutions%
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 |