2008 USAMO Problems/Problem 2
Contents
- 1 Problem
- 2 Solutions
- 2.1 Solution 1 (synthetic)
- 2.2 Solution 2 (complex)
- 2.3 Solution 3 (synthetic)
- 2.4 Solution 4 (synthetic)
- 2.5 Solution 5 (trigonometric)
- 2.6 Solution 6 (isogonal conjugates)
- 2.7 Solution 7 (symmedians)
- 2.8 Solution 8 (inversion) (Official Solution #2)
- 2.9 Solution 9 (Official Solution #1)
- 2.10 Solution 10 (Anti-Steiner point)
- 3 See Also
Problem
(Zuming Feng) Let be an acute, scalene triangle, and let , , and be the midpoints of , , and , respectively. Let the perpendicular bisectors of and intersect ray in points and respectively, and let lines and intersect in point , inside of triangle . Prove that points , , , and all lie on one circle.
Solutions
Solution 1 (synthetic)
Without Loss of Generality, assume . It is sufficient to prove that , as this would immediately prove that are concyclic. By applying the Menelaus' Theorem in the Triangle for the transversal , we have (in magnitude) Here, we used that , as is the midpoint of . Now, since and , we have Now, note that bisects the exterior and bisects exterior , making the -excentre of . This implies that bisects interior , making , as was required.
Solution 2 (complex)
Let where all lie on the unit circle. Then is 0. As noted earlier, is cyclic. We will find the ghost point the second intersection of and .
We know that these two circles already intersect at so we can reflect over the line between their centers. The center of is the midpoint of namely . With the tangent formula and then taking the midpoint, we find that the center of is Then we want to find the reflection of 0 over the line through and Then we get
Now it remains to show the other angle equality would follow by symmetry.
Then we get: Thus so and we're done.
~cocohearts
Solution 3 (synthetic)
Without loss of generality . The intersection of and is , the circumcenter of .
Let and . Note lies on the perpendicular bisector of , so . So . Similarly, , so . Notice that intercepts the minor arc in the circumcircle of , which is double . Hence , so is cyclic.
Lemma. is directly similar to
Proof. since , , are collinear, is cyclic, and . Also because , and is the medial triangle of so . Hence .
Notice that since . . Then Hence .
Hence is similar to by AA similarity. It is easy to see that they are oriented such that they are directly similar.
End Lemma
By the similarity in the Lemma, . so by SAS similarity. Hence Using essentially the same angle chasing, we can show that is directly similar to . It follows that is directly similar to . So Hence , so is cyclic. In other words, lies on the circumcircle of . Note that , so is cyclic. In other words, lies on the circumcircle of . , , , , and all lie on the circumcircle of . Hence , , , and lie on a circle, as desired.
Solution 4 (synthetic)
This solution utilizes the phantom point method. Clearly, APON are cyclic because . Let the circumcircles of triangles and intersect at and .
Lemma. If are points on circle with center , and the tangents to at intersect at , then is the symmedian from to .
Proof. This is fairly easy to prove (as H, O are isogonal conjugates, plus using SAS similarity), but the author lacks time to write it up fully, and will do so soon.
End Lemma
It is easy to see (the intersection of ray and the circumcircle of ) is colinear with and , and because line is the diameter of that circle, , so is the point in the lemma; hence, we may apply the lemma. From here, it is simple angle-chasing to show that satisfies the original construction for , showing ; we are done.
Solution 5 (trigonometric)
By the Law of Sines, . Since and similarly , we cancel to get . Obviously, so .
Then and . Subtracting these two equations, so . Therefore, (by AA similarity), so a spiral similarity centered at takes to and to . Therefore, it takes the midpoint of to the midpoint of , or to . So and is cyclic.
Solution 6 (isogonal conjugates)
Construct on such that . Then . Then , so , or . Then , so . Then we have
and . So and are isogonally conjugate. Thus . Then
.
If is the circumcenter of then so is cyclic. Then .
Then . Then is a right triangle.
Now by the homothety centered at with ratio , is taken to and is taken to . Thus is taken to the circumcenter of and is the midpoint of , which is also the circumcenter of , so all lie on a circle.
Solution 7 (symmedians)
Median of a triangle implies . Trig ceva for shows that is a symmedian. Then is a median, use the lemma again to show that , and similarly , so you're done.
Solution 8 (inversion) (Official Solution #2)
Invert the figure about a circle centered at , and let denote the image of the point under this inversion. Find point so that is a parallelogram and let denote the center of this parallelogram. Note that and . Because is the midpoint of and is the midpoint of , we also have . Thus Hence quadrilateral is cyclic and, by a similar argument, quadrilateral is also cyclic. Because the images under the inversion of lines and are circles that intersect in and , it follows that .
Next note that , , and are collinear and are the images of , , and , respectively, under a homothety centered at and with ratio . It follows that , , and are collinear, and then that the points , , , and lie on a circle.
Solution 9 (Official Solution #1)
Let be the circumcenter of triangle . We prove that It will then follow that lie on the circle with diameter . Indeed, the fact that the first two angles in are right is immediate because and are the perpendicular bisectors of and , respectively. Thus we need only prove that .
We may assume, without loss of generality, that . This leads to configurations similar to the ones shown above. The proof can be adapted to other configurations. Because is the perpendicular bisector of , it follows that triangle is an isosceles triangle with . Likewise, triangle is isosceles with . Let and , so .
Applying the Law of Sines to triangles and gives Taking the quotient of the two equations and noting that , we find Because , we have Applying the Law of Sines to triangles and , we find Taking the quotient of the two equations yields so by , Because is an exterior angle to triangle , we have . Similarly, . Hence Thus , so is cyclic. In addition, and hence, from , . Because is cyclic and is isosceles with vertex angle , we have . Therefore, This completes the proof.
Solution 10 (Anti-Steiner point)
First, let be the circumcenter of . Note that since and , then , , and , are concyclic.
Notice that and , which means . It follows analogously that . This means that is the orthocenter of triangle .
Now consider what happens when we reflect line over line . Since is just the perpendicular bisector of , point will map to point , and point , which is both line and , will map to itself. Therefore, we get line from this reflection. It follows analogously that by reflecting line over line , we get .
Since line passes through , the orthocenter of triangle , its reflections over sides and , which correspond to and respectively, intersect on (this is the Anti-Steiner point application). Therefore, , the intersection of and , lies on .
Since , , , and are concyclic, then is the same as , so lies on , which is the same as saying , , , and are concyclic, as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
- <url>viewtopic.php?t=202907 Discussion on AoPS/MathLinks</url>
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