Difference between revisions of "2010 IMO Problems/Problem 2"
m (→Solution) |
(→Solution) |
||
Line 6: | Line 6: | ||
== Solution == | == Solution == | ||
− | Note that it suffices to prove alternatively that if <math>EI</math> meets the circle again at <math>J</math> and <math>JD</math> meets <math>IF</math> at <math>G</math>, then <math>G</math> is the midpoint of <math>IF</math>. | + | Note that it suffices to prove alternatively that if <math>EI</math> meets the circle again at <math>J</math> and <math>JD</math> meets <math>IF</math> at <math>G</math>, then <math>G</math> is the midpoint of <math>IF</math>. Let <math>JD</math> meet <math>BC</math> at <math>K</math>. |
Observation 1. D is the midpoint of arc <math>BDC</math> because it lies on angle bisector <math>AI</math>. | Observation 1. D is the midpoint of arc <math>BDC</math> because it lies on angle bisector <math>AI</math>. |
Revision as of 18:50, 11 November 2015
Problem
Given a triangle , with
as its incenter and
as its circumcircle,
intersects
again at
. Let
be a point on arc
, and
a point on the segment
, such that
. If
is the midpoint of
, prove that the intersection of lines
and
lies on
.
Authors: Tai Wai Ming and Wang Chongli, Hong Kong
Solution
Note that it suffices to prove alternatively that if meets the circle again at
and
meets
at
, then
is the midpoint of
. Let
meet
at
.
Observation 1. D is the midpoint of arc because it lies on angle bisector
.
Observation 2. bisects
as well.
Key Lemma. Triangles and
are similar.
Proof. Because triangles
and
are similar by AA Similarity (for
and
both intercept equally sized arcs), we have
. But we know that triangle
is isosceles (hint: prove
), and so
. Hence, by SAS Similarity, triangles
and
are similar, as desired.
Observation 3. As a result, we have .
Observation 4. .
Observation 5. If and
intersect at
, then
is cyclic.
Observation 6. Because , we have
.
Observation 7. is a parallelogram, so its diagonals bisect each other, so
is the midpoint of
, as desired.
See Also
2010 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |