Difference between revisions of "2015 USAJMO Problems/Problem 3"
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===Solution 3=== | ===Solution 3=== | ||
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Note that each point <math>X</math> on <math>PQ</math> corresponds to exactly one point on arc <math>PBQ</math>. Also notice that since <math>AB</math> is the diameter of <math>\omega</math>, <math>\angle ASB</math> is always a right angle; therefore, point <math>T</math> is always <math>B</math>. WLOG, assume that <math>\omega</math> is on the coordinate plane, and <math>B</math> corresponds to the origin. The locus of <math>M</math>, since the locus of <math>S</math> is arc <math>PBQ</math>, is the arc that is produced when arc <math>PBQ</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin, which resides on the circle <math>\psi</math>, which is produced when <math>\omega</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin. By MSmathlete1018 | Note that each point <math>X</math> on <math>PQ</math> corresponds to exactly one point on arc <math>PBQ</math>. Also notice that since <math>AB</math> is the diameter of <math>\omega</math>, <math>\angle ASB</math> is always a right angle; therefore, point <math>T</math> is always <math>B</math>. WLOG, assume that <math>\omega</math> is on the coordinate plane, and <math>B</math> corresponds to the origin. The locus of <math>M</math>, since the locus of <math>S</math> is arc <math>PBQ</math>, is the arc that is produced when arc <math>PBQ</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin, which resides on the circle <math>\psi</math>, which is produced when <math>\omega</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin. By MSmathlete1018 | ||
The above solution is false; it gives the wrong locus. The step where they mess up is in stating that <math>T=B</math>, however this is not true because <math>\angle TXA</math> is right, not <math>\angle TSA</math>. | The above solution is false; it gives the wrong locus. The step where they mess up is in stating that <math>T=B</math>, however this is not true because <math>\angle TXA</math> is right, not <math>\angle TSA</math>. |
Revision as of 10:53, 16 April 2017
Contents
Problem
Quadrilateral is inscribed in circle
with
and
. Let
be a variable point on segment
. Line
meets
again at
(other than
). Point
lies on arc
of
such that
is perpendicular to
. Let
denote the midpoint of chord
. As
varies on segment
, show that
moves along a circle.
Solution 1
We will use coordinate geometry.
Without loss of generality,
let the circle be the unit circle centered at the origin,
,
where
.
Let angle , which is an acute angle,
, then
.
Angle ,
.
Let
, then
.
The condition yields:
(E1)
Use identities ,
,
, we obtain
. (E1')
The condition that is on the circle yields
, namely
. (E2)
is the mid-point on the hypotenuse of triangle
, hence
, yielding
. (E3)
Expand (E3), using (E2) to replace with
, and using (E1') to replace
with
, and we obtain
, namely
, which is a circle centered at
with radius
.
Solution 2
Let the midpoint of be
. We claim that
moves along a circle with radius
.
We will show that , which implies that
, and as
is fixed, this implies the claim.
by the median formula on
.
by the median formula on
.
.
As ,
from right triangle
.
By ,
.
Since is the circumcenter of
, and
is the circumradius, the expression
is the power of point
with respect to
. However, as
is also the power of point
with respect to
, this implies that
.
By ,
Finally, by AA similarity (
and
), so
.
By ,
, so
, as desired.
Solution 3
The below solution is false.
Note that each point on
corresponds to exactly one point on arc
. Also notice that since
is the diameter of
,
is always a right angle; therefore, point
is always
. WLOG, assume that
is on the coordinate plane, and
corresponds to the origin. The locus of
, since the locus of
is arc
, is the arc that is produced when arc
is dilated by
with respect to the origin, which resides on the circle
, which is produced when
is dilated by
with respect to the origin. By MSmathlete1018
The above solution is false; it gives the wrong locus. The step where they mess up is in stating that , however this is not true because
is right, not
.