Difference between revisions of "2007 IMO Problems/Problem 4"
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<math>(tkhalid)</math> | <math>(tkhalid)</math> | ||
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+ | ==Solution 2 (Power of a point)== | ||
+ | <math>\angle{RQL}=90+\angle{QCL}=90+\dfrac{C}{2}</math>, and similarly <math>\angle{RPK}=90+\angle{PCK}=90+\dfrac{C}{2}</math>, we have <math>\angle{RQL}=\angle{RPK}</math>. Using triangle area formula <math>A=bc\sin{\angle{A}}</math>, the problem is equivalent to proving <math>RQ*QL=RP*PK</math>, or <math>\dfrac{PK}{QL}=\dfrac{RQ}{RP}</math>. Draw line <math>QM</math> perpendicular to BC and intersects BC at <math>M</math>, then <math>QM=QL</math>, and <math>\dfrac{PC}{QC}=\dfrac{PK}{QM}=\dfrac{PK}{QL}</math>. Now the problem is equivalent to proving <math>\dfrac{PC}{QC}=\dfrac{RQ}{RP}</math>, or <math>RQ*QC=RP*PC</math>. Since <math>\angle{OPQ}=180-\angle{RPK}=180-\angle{RQL}=\angle{OQP}</math>, we have <math>OQ=OP=x</math>. Let the radius of the circumcircle be <math>r</math>, then the diameter through <math>P</math> is divided by point <math>P</math> into lengths of <math>r+x</math> and <math>r-x</math>. By power of point, <math>RP*PC=(r+x)(r-x)</math>. Similarly, <math>RQ*QC=(r+x)(r-x)</math>. Therefore <math>RP*PC=RQ*QC</math>. <math>\square</math> | ||
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+ | <math>(mathdummy)</math> | ||
{{alternate solutions}} | {{alternate solutions}} |
Revision as of 02:31, 19 August 2018
Problem
In the bisector of
intersects the circumcircle again at
, the perpendicular bisector of
at
, and the perpendicular bisector of
at
. The midpoint of
is
and the midpoint of
is
. Prove that the triangles
and
have the same area.
Solution
The area of is given by
and the area of
is
. Let
,
, and
. Now
and
, thus
.
, so
, or
. The ratio of the areas is
. The two areas are only equal when the ratio is 1, therefore it suffices to show
. Let
be the center of the circle. Then
, and
. Using law of sines on
we have:
so
.
by law of sines, and
, thus 1)
. Similarly, law of sines on
results in
or
. Cross multiplying we have
or 2)
. Dividing 1) by 2) we have
Solution 2 (Power of a point)
, and similarly
, we have
. Using triangle area formula
, the problem is equivalent to proving
, or
. Draw line
perpendicular to BC and intersects BC at
, then
, and
. Now the problem is equivalent to proving
, or
. Since
, we have
. Let the radius of the circumcircle be
, then the diameter through
is divided by point
into lengths of
and
. By power of point,
. Similarly,
. Therefore
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
2007 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |