Difference between revisions of "2010 IMO Problems/Problem 4"
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Let the tangent at <math>M</math> to <math>\Gamma</math> intersect <math>SC</math> at <math>X</math>. We now have that since <math>\triangle{XMC}</math> and <math>\triangle{SPC}</math> are both isosceles, <math>\angle{SPC}=\angle{SCP}=\angle{XMC}</math>. This yields that <math>MX \| PS</math>. | Let the tangent at <math>M</math> to <math>\Gamma</math> intersect <math>SC</math> at <math>X</math>. We now have that since <math>\triangle{XMC}</math> and <math>\triangle{SPC}</math> are both isosceles, <math>\angle{SPC}=\angle{SCP}=\angle{XMC}</math>. This yields that <math>MX \| PS</math>. |
Revision as of 00:08, 16 July 2010
Problem
Let be a point interior to triangle
(with
). The lines
,
and
meet again its circumcircle
at
,
, respectively
. The tangent line at
to
meets the line
at
. Show that from
follows
.
Solution
Solution 1
Without loss of generality, suppose that . By Power of a Point,
, so
is tangent to the circumcircle of
. Thus,
. It follows that after some angle-chasing,
so
as desired.
Solution 2
Let the tangent at to
intersect
at
. We now have that since
and
are both isosceles,
. This yields that
.
Now consider the power of point with respect to
.
Hence by AA similarity, we have that . Combining this with the arc angle theorem yields that
. Hence
.
This implies that the tangent at is parallel to
and therefore that
is the midpoint of arc
. Hence
.
See also
2010 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |