Difference between revisions of "2016 USAMO Problems/Problem 3"
(Created page with "==Problem== Let <math>\triangle ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math>-excenter, <math>C</math>-excenter,...") |
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Lines <math>I_B F</math> and <math>I_C E</math> meet at <math>P.</math> Prove that <math>\overline{PO}</math> and <math>\overline{YZ}</math> are perpendicular. | Lines <math>I_B F</math> and <math>I_C E</math> meet at <math>P.</math> Prove that <math>\overline{PO}</math> and <math>\overline{YZ}</math> are perpendicular. | ||
==Solution== | ==Solution== | ||
− | {{ | + | There are two major steps of a proof. |
+ | |||
+ | 1. Let <math>I_A</math> be the <math>A</math>-excenter, then <math>I_A,O,P</math> are colinear. This can be proved by the Trigonometric Form of Ceva's Theorem for <math>triangle I_AI_BI_C.</math> | ||
+ | |||
+ | 2. Show that <math>I_AY^2-I_AZ^2=OY^2-OZ^2,</math> which shows <math>\overline{OI_A}\perp\overline{YZ}.</math> This can be proved by multiple applications of the Pythagorean Thm. | ||
{{MAA Notice}} | {{MAA Notice}} | ||
==See also== | ==See also== | ||
{{USAMO newbox|year=2016|num-b=2|num-a=4}} | {{USAMO newbox|year=2016|num-b=2|num-a=4}} |
Revision as of 14:24, 17 February 2017
Problem
Let be an acute triangle, and let
and
denote its
-excenter,
-excenter, and circumcenter, respectively. Points
and
are selected on
such that
and
Similarly, points
and
are selected on
such that
and
Lines and
meet at
Prove that
and
are perpendicular.
Solution
There are two major steps of a proof.
1. Let be the
-excenter, then
are colinear. This can be proved by the Trigonometric Form of Ceva's Theorem for
2. Show that which shows
This can be proved by multiple applications of the Pythagorean Thm.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
See also
2016 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |