Difference between revisions of "2006 IMO Problems/Problem 1"
Smileapple (talk | contribs) m |
Smileapple (talk | contribs) m |
||
Line 11: | Line 11: | ||
By (1), (2), and (3), we get <math>\angle IBP = \angle ICP</math>; hence <math>B,I,P,C</math> are concyclic. | By (1), (2), and (3), we get <math>\angle IBP = \angle ICP</math>; hence <math>B,I,P,C</math> are concyclic. | ||
− | Let ray <math>AI</math> meet the circumcircle of <math>\triangle | + | Let ray <math>AI</math> meet the circumcircle of <math>\triangle ABC</math> at point <math>J</math>. Then, by the Incenter-Excenter Lemma, <math>JB=JC=JI=JP</math>. |
Finally, <math>AP+JP \geq AJ = AI+IJ</math> (since triangle APJ can be degenerate, which happens only when <math>P=I</math>), but <math>JI=JP</math>; hence <math>AP \geq AI</math> and we are done. | Finally, <math>AP+JP \geq AJ = AI+IJ</math> (since triangle APJ can be degenerate, which happens only when <math>P=I</math>), but <math>JI=JP</math>; hence <math>AP \geq AI</math> and we are done. |
Revision as of 05:06, 23 December 2022
Problem
Let be triangle with incenter
. A point
in the interior of the triangle satisfies
. Show that
, and that equality holds if and only if
Solution
We have
(1)
and similarly
(2).
Since
, we have
(3).
By (1), (2), and (3), we get ; hence
are concyclic.
Let ray meet the circumcircle of
at point
. Then, by the Incenter-Excenter Lemma,
.
Finally, (since triangle APJ can be degenerate, which happens only when
), but
; hence
and we are done.
By Mengsay LOEM , Cambodia IMO Team 2015
latexed by tluo5458 :)