Difference between revisions of "2006 IMO Problems/Problem 1"
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==Problem== | ==Problem== | ||
− | Let <math>ABC</math> be triangle with incenter <math>I</math>. A point <math>P</math> in the interior of the triangle satisfies <math>\angle PBA+\angle PCA = \angle PBC+\angle PCB</math>. Show that <math>AP \geq AI</math>, and that equality holds if and only if <math>P=I</math> | + | Let <math>ABC</math> be triangle with incenter <math>I</math>. A point <math>P</math> in the interior of the triangle satisfies <math>\angle PBA+\angle PCA = \angle PBC+\angle PCB</math>. Show that <math>AP \geq AI</math>, and that equality holds if and only if <math>P=I.</math> |
− | ==Solution==. | + | ==Solution== |
+ | We have | ||
+ | <cmath>\angle IBP = \angle IBC - \angle PBC = \frac{1}{2} \angle ABC - \angle PBC = \frac{1}{2}(\angle PCB - \angle PCA).</cmath> | ||
+ | |||
+ | and similarly <cmath>\angle ICP = \angle PCB - \angle ICB = \angle PCB - \frac{1}{2} \angle ACB = \frac{1}{2}(\angle PBA - \angle PBC).</cmath> | ||
+ | Since <math>\angle PBA + \angle PCA = \angle PBC + \angle PCB</math>, we have <math>\angle PCB - \angle PCA = \angle PBA - \angle PBC.</math> | ||
+ | |||
+ | It follows that <cmath>\angle IBP = \frac{1}{2} (\angle PCB - \angle PCA) = \frac{1}{2} (\angle PBA - \angle PBC) = \angle ICP.</cmath> Hence, <math>B,P,I,</math> and <math>C</math> are concyclic. | ||
+ | |||
+ | |||
+ | 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. | ||
+ | |||
+ | |||
+ | By Mengsay LOEM , Cambodia IMO Team 2015 | ||
+ | |||
+ | latexed by tluo5458 :) | ||
+ | |||
+ | minor edits by lpieleanu | ||
+ | |||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2006|before=First Problem|num-a=2}} |
Latest revision as of 01:02, 19 November 2023
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
and similarly
Since
, we have
It follows that Hence,
and
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 :)
minor edits by lpieleanu
See Also
2006 IMO (Problems) • Resources | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |