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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==
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{{solution}}
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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 13:24, 17 February 2017

Problem

Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY = \angle CBY$ and $\overline{BE}\perp\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ = \angle BCZ$ and $\overline{CF}\perp\overline{AB}.$

Lines $I_B F$ and $I_C E$ meet at $P.$ Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular.

Solution

There are two major steps of a proof.

1. Let $I_A$ be the $A$-excenter, then $I_A,O,P$ are colinear. This can be proved by the Trigonometric Form of Ceva's Theorem for $triangle I_AI_BI_C.$

2. Show that $I_AY^2-I_AZ^2=OY^2-OZ^2,$ which shows $\overline{OI_A}\perp\overline{YZ}.$ 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. AMC logo.png

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
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