Difference between revisions of "1993 USAMO Problems/Problem 2"
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Let <math>ABCD</math> be a convex quadrilateral such that diagonals <math>AC</math> and <math>BD</math> intersect at right angles, and let <math>E</math> be their intersection. Prove that the reflections of <math>E</math> across <math>AB</math>, <math>BC</math>, <math>CD</math>, <math>DA</math> are concyclic. | Let <math>ABCD</math> be a convex quadrilateral such that diagonals <math>AC</math> and <math>BD</math> intersect at right angles, and let <math>E</math> be their intersection. Prove that the reflections of <math>E</math> across <math>AB</math>, <math>BC</math>, <math>CD</math>, <math>DA</math> are concyclic. | ||
− | == Solution == | + | == Solution 1 == |
===Diagram=== | ===Diagram=== | ||
<center><table border=1><tr><td><asy> | <center><table border=1><tr><td><asy> | ||
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===Work=== | ===Work=== | ||
− | Let <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> be the foot of the | + | Let <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> be the foot of the altitude from point <math>E</math> of <math>\triangle AEB</math>, <math>\triangle BEC</math>, <math>\triangle CED</math>, <math>\triangle DEA</math>. |
− | Note that reflection of <math>E</math> over the | + | Note that reflection of <math>E</math> over all the points of <math>XYZW</math> is similar to <math>XYZW</math> with a scale of <math>2</math> with center <math>E</math>. Thus, if <math>XYZW</math> is cyclic, then the reflections are cyclic. |
<br/> | <br/> | ||
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<P align="right"><math>\mathbb{Q.E.D}</math></P> | <P align="right"><math>\mathbb{Q.E.D}</math></P> | ||
+ | |||
+ | == Solution 2 == | ||
+ | |||
+ | Suppose the reflection of E over AB is W, and similarly define X, Y, and Z. | ||
+ | <math>\bigtriangleup BEA \cong \bigtriangleup BWA</math> by reflection gives <math>BE = BW</math> | ||
+ | <math>\bigtriangleup BEC \cong \bigtriangleup BXC</math> by reflection gives <math>BE = BX</math> | ||
+ | These two tell us that E, W, and X belong to a circle with center B. | ||
+ | Similarly, we can get that: | ||
+ | E, Z, and W belong to a circle with center A, | ||
+ | E, X, and Y belong to a circle with center C, | ||
+ | E, Y, and Z belong to a circle with center D. | ||
+ | |||
+ | To prove that W, X, Y, Z are concyclic, we want to prove <math>\angle XWZ + \angle XYZ = 180^o</math> | ||
+ | <math>\angle XWZ + \angle XYZ = \angle XWE + \angle EWZ + \angle XYE + \angle EYZ</math> | ||
+ | <math> = \frac{1}{2} \angle XBE + \frac{1}{2} \angle EAZ + \frac{1}{2} \angle XCE + \frac{1}{2} \angle EDZ</math> | ||
+ | <math> = \frac{1}{2} (\angle XBE + \angle XCE) + \frac{1}{2} (\angle EAZ + \angle EDZ)</math> | ||
+ | |||
+ | <math>\angle AED = 90^o</math> and <math>\angle AED = \angle AZD</math> tells us that <math>\angle EAZ + \angle EDZ = 180^o</math> | ||
+ | Similarly, <math>\angle XBE + \angle XCE = 180^o</math> | ||
+ | Thus, <math>\angle XWZ + \angle XYZ = \frac{1}{2} \cdot 180^o + \frac{1}{2} \cdot 180^o = 180^o</math>, and we are done. | ||
+ | -- Lucas.xue (someone pls help with a diagram) | ||
+ | |||
== See Also == | == See Also == |
Latest revision as of 20:26, 5 September 2024
Problem 2
Let be a convex quadrilateral such that diagonals
and
intersect at right angles, and let
be their intersection. Prove that the reflections of
across
,
,
,
are concyclic.
Solution 1
Diagram
![]() |
Work
Let ,
,
,
be the foot of the altitude from point
of
,
,
,
.
Note that reflection of over all the points of
is similar to
with a scale of
with center
. Thus, if
is cyclic, then the reflections are cyclic.
is right angle and so is
. Thus,
is cyclic with
being the diameter of the circumcircle.
Follow that, because they inscribe the same angle.
Similarly ,
,
.
Futhermore, .
Thus, and
are supplementary and follows that,
is cyclic.
Solution 2
Suppose the reflection of E over AB is W, and similarly define X, Y, and Z.
by reflection gives
by reflection gives
These two tell us that E, W, and X belong to a circle with center B.
Similarly, we can get that:
E, Z, and W belong to a circle with center A,
E, X, and Y belong to a circle with center C,
E, Y, and Z belong to a circle with center D.
To prove that W, X, Y, Z are concyclic, we want to prove
and
tells us that
Similarly,
Thus,
, and we are done.
-- Lucas.xue (someone pls help with a diagram)
See Also
1993 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.