Difference between revisions of "2012 USAJMO Problems/Problem 1"
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==Problem== | ==Problem== | ||
− | Given a triangle <math>ABC</math>, let <math>P</math> and <math>Q</math> be points on segments <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, such that <math>AP = AQ</math>. Let <math>S</math> and <math>R</math> be distinct points on segment <math>\overline{BC}</math> such that <math>S</math> lies between <math>B</math> and <math>R</math>, <math>\angle BPS = \angle PRS</math>, and <math>\angle CQR = \angle QSR</math>. Prove that <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math> are concyclic (in other words, these four points lie on a circle). | + | Given a triangle <math>ABC</math>, let <math>P</math> and <math>Q</math> be points on segments <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, such that <math>AP = AQ</math>. Let <math>S</math> and <math>R</math> be distinct points on segment <math>\overline{BC}</math> such that <math>S</math> lies between <math>B</math> and <math>R</math>, <math>\angle{BPS} = \angle{PRS}</math>, and <math>\angle{CQR} = \angle{QSR}</math>. Prove that <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math> are concyclic (in other words, these four points lie on a circle). |
==Solution== | ==Solution== | ||
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</asy> | </asy> | ||
− | For the sake of contradiction, suppose that the circumcircles of triangles <math>PRS</math> and <math>QRS</math> are not the same circle. Since <math>AP = AQ</math>, <math>A</math> lies on the [[Radical_axis|radical axis]] of both circles. However, both circles pass through <math>R</math> and <math>S</math>, so the radical axis of both circles is <math>RS</math>. Hence, <math>A</math> lies on <math>RS</math>, which is a contradiction. | + | For the sake of contradiction, suppose that the circumcircles of triangles <math>PRS</math> and <math>QRS</math> are not the same circle. Since <math>AP = AQ</math>, <math>AP^2 = AQ^2</math>, so the power of A with respect to both circumcircles is the same. Thus, <math>A</math> lies on the [[Radical_axis|radical axis]] of both circles. However, both circles pass through <math>R</math> and <math>S</math>, so the radical axis of both circles is <math>RS</math>. Hence, <math>A</math> lies on <math>RS</math>, which is a contradiction. |
Therefore, the two circumcircles are the same circle. In other words, <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> all lie on the same circle. | Therefore, the two circumcircles are the same circle. In other words, <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> all lie on the same circle. |
Latest revision as of 20:50, 19 December 2023
Contents
Problem
Given a triangle , let
and
be points on segments
and
, respectively, such that
. Let
and
be distinct points on segment
such that
lies between
and
,
, and
. Prove that
,
,
,
are concyclic (in other words, these four points lie on a circle).
Solution
Since , the circumcircle of triangle
is tangent to
at
. Similarly, since
, the circumcircle of triangle
is tangent to
at
.
For the sake of contradiction, suppose that the circumcircles of triangles and
are not the same circle. Since
,
, so the power of A with respect to both circumcircles is the same. Thus,
lies on the radical axis of both circles. However, both circles pass through
and
, so the radical axis of both circles is
. Hence,
lies on
, which is a contradiction.
Therefore, the two circumcircles are the same circle. In other words, ,
,
, and
all lie on the same circle.
Solution 2
Note that (as in the first solution) the circumcircle of triangle is tangent to
at
. Similarly, since
, the circumcircle of triangle
is tangent to
at
.
Now, suppose these circumcircles are not the same circle. They already intersect at and
, so they cannot intersect anymore. Thus, AS must touch the two circumcircles at points
and
, with
on the circumcircle of triangle
. By Power of a Point,
and
. Hence, because
,
, a contradiction because then, as they lie on the same line segment, M and N must be the same point! (Note line segment, not line.) Hence, the two circumcircles are the same circle.
See also
2012 USAJMO (Problems • Resources) | ||
First Problem | Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.