Difference between revisions of "2014 USAMO Problems/Problem 5"
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Since <math>AHPC</math> is a cyclic quadrilateral, <math>\angle AHC = \angle APC</math>. <math>\angle AHC = 90^\circ + \angle ABC</math> and <math>\angle APC = 90^\circ + \angle AYC</math>, we find <math>\angle ABC = \angle AYC</math>. That is, <math>ABYC</math> is a cyclic quadrilateral. Let <math>D</math> be mid-point of <math>\overline{AB}</math>. <math>O, X, D</math> are collinear and <math>OX \perp AB</math>. Let <math>M</math> be second intersection of <math>AP</math> with circumcircle of the triangle <math>ABC</math>. Let <math>YP \cap AC = E</math>, <math>YM \cap AB = F</math>. Since <math>M</math> is mid-point of the arc <math>BC</math>, <math>OM\perp BC</math>. Since <math>AYMC</math> is a cyclic quadrilateral, <math>\angle CYM = \angle CAM = \angle BAC /2</math>. Since <math>Y</math> is the orthocenter of triangle <math>APC</math>, <math>\angle PYC = \angle CAP = \angle BAC /2</math>. Thus, <math>\angle PYM = \angle BAC</math> and <math>AEYF</math> is a cyclic quadrilateral. So, <math>YF \perp AB</math> and <math>OX \parallel MY</math>. We will prove that <math>XYMO</math> is a parallelogram. | Since <math>AHPC</math> is a cyclic quadrilateral, <math>\angle AHC = \angle APC</math>. <math>\angle AHC = 90^\circ + \angle ABC</math> and <math>\angle APC = 90^\circ + \angle AYC</math>, we find <math>\angle ABC = \angle AYC</math>. That is, <math>ABYC</math> is a cyclic quadrilateral. Let <math>D</math> be mid-point of <math>\overline{AB}</math>. <math>O, X, D</math> are collinear and <math>OX \perp AB</math>. Let <math>M</math> be second intersection of <math>AP</math> with circumcircle of the triangle <math>ABC</math>. Let <math>YP \cap AC = E</math>, <math>YM \cap AB = F</math>. Since <math>M</math> is mid-point of the arc <math>BC</math>, <math>OM\perp BC</math>. Since <math>AYMC</math> is a cyclic quadrilateral, <math>\angle CYM = \angle CAM = \angle BAC /2</math>. Since <math>Y</math> is the orthocenter of triangle <math>APC</math>, <math>\angle PYC = \angle CAP = \angle BAC /2</math>. Thus, <math>\angle PYM = \angle BAC</math> and <math>AEYF</math> is a cyclic quadrilateral. So, <math>YF \perp AB</math> and <math>OX \parallel MY</math>. We will prove that <math>XYMO</math> is a parallelogram. | ||
− | + | https://wiki-images.artofproblemsolving.com//7/7b/Usamo2014-5.png | |
We see that <math>YPM</math> is an isosceles triangle and <math>YM=YP</math>. Also <math>XB=XP</math> and <math>\angle BXP = 2\angle BAP = \angle BAC = \angle PYM</math>. Then, <math> BXP \sim MYP </math>. By spiral similarity, <math> BPM \sim XPY </math> and <math>\angle XYP = \angle BMP = \angle BCA</math>. Hence, <math>\angle XYP = \angle BCA</math>, <math>XY \perp BC</math>. Since <math>OM \perp BC</math>, we get <math>XYMO</math> is a parallelogram. As a result, <math>OM = XY</math>. | We see that <math>YPM</math> is an isosceles triangle and <math>YM=YP</math>. Also <math>XB=XP</math> and <math>\angle BXP = 2\angle BAP = \angle BAC = \angle PYM</math>. Then, <math> BXP \sim MYP </math>. By spiral similarity, <math> BPM \sim XPY </math> and <math>\angle XYP = \angle BMP = \angle BCA</math>. Hence, <math>\angle XYP = \angle BCA</math>, <math>XY \perp BC</math>. Since <math>OM \perp BC</math>, we get <math>XYMO</math> is a parallelogram. As a result, <math>OM = XY</math>. |
Revision as of 11:06, 27 March 2022
Contents
Problem
Let be a triangle with orthocenter and let be the second intersection of the circumcircle of triangle with the internal bisector of the angle . Let be the circumcenter of triangle and the orthocenter of triangle . Prove that the length of segment is equal to the circumradius of triangle .
Solution 1
Let be the center of , be the center of . Note that is the reflection of across , so . Additionally so lies on . Now since are perpendicular to and their bisector, is isosceles with , and . Also But as well, and , so . Thus .
Solution 2
Since is a cyclic quadrilateral, . and , we find . That is, is a cyclic quadrilateral. Let be mid-point of . are collinear and . Let be second intersection of with circumcircle of the triangle . Let , . Since is mid-point of the arc , . Since is a cyclic quadrilateral, . Since is the orthocenter of triangle , . Thus, and is a cyclic quadrilateral. So, and . We will prove that is a parallelogram.
https://wiki-images.artofproblemsolving.com//7/7b/Usamo2014-5.png
We see that is an isosceles triangle and . Also and . Then, . By spiral similarity, and . Hence, , . Since , we get is a parallelogram. As a result, . (Lokman GÖKÇE)
See also
2014 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |