2014 IMO Problems/Problem 4
Points and lie on side of acute-angled so that and . Points and lie on lines and , respectively, such that is the midpoint of , and is the midpoint of . Prove that lines and intersect on the circumcircle of .
We are trying to prove that the intersection of and , call it point , is on the circumcircle of triangle . In other words, we are trying to prove . Let the intersection of and be point , and the intersection of and be point . Let us assume . Note: This is circular reasoning. If , then should be equal to and . We can quickly prove that the triangles , , and are similar, so . We also see that . Also because angles and and are equal, the triangles and , and must be two pairs of similar triangles. Therefore we must prove angles and and are equal. We have angles . We also have , . Because the triangles and are similar, we have , so triangles and are similar. So the angles and and are equal and we are done.
Let be the midpoint of . Easy angle chasing gives . Because is the midpoint of , the cotangent rule applied on triangle gives us Hence, by the cotangent rule on , we have Because the period of cotangent is , but angles are less than , we have
Similarly, we have Hence, if and intersect at , then by the Angle Sum in a Triangle Theorem. Hence, is cyclic, which is equivalent to the desired result.
--Suli 23:27, 7 February 2015 (EST)
Let be the midpoint of . By AA Similarity, triangles and are similar, so and . Similarly, , and so triangle is isosceles. Thus, , and so . Dividing both sides by 2, we have , or But we also have , so triangles and are similar by similarity. In particular, . Similarly, , so . In addition, angle sum in triangle gives . Therefore, if we let lines and intersect at , by Angle Sum in quadrilateral concave , and so convex , which is enough to prove that is cyclic. This completes the proof.
[asy] size(250); defaultpen(fontsize(8pt));
pair A = dir(110); pair B = dir(210); pair C = dir(330); pair Pp = rotate(50, A)*B; pair P = extension(A,Pp,B,C); pair Qp = rotate(-70, A)*C; pair Q = extension(A,Qp,B,C); pair M = rotate(180,P)*A; pair N = rotate(180,Q)*A; path c1 = circumcircle(A,B,C); pair T = IP(B--M,C--N); pair L = midpoint(B--C);
dot("", A, dir(100)); dot("", B, dir(-110)); dot("", C, dir(-40)); dot("", P, dir(50)); dot("", Q, dir(-170)); dot("", M, dir(-50)); dot("", N, dir(-140)); dot("", T,dir(-90)); dot("", L, dir(-120)); [/asy]
--Suli 10:38, 8 February 2015 (EST)
Let be the second intersection of with the circumcircle of and the second intersection of with the circumcircle of By inscribed angles, the tangent at is parallel to Let denote the point at infinity along line Note that So, is harmonic. Similarly, we can find is harmonic. Therefore, which means that and intersect on the circumcircle.
We use barycentric coordinates. Due to the equal angles, is tangent to the circumcircle of and is tangent to the circumcircle of the Therefore, we can use power of a point to solve for side ratios. We have Therefore, as and are cevians. Note that lies on the circumcircle iff Substituting the values in, we have so we are done.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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