MathLabs

Problem 3

Let ABCABC be a scalene triangle with circumcircle Ω\Omega. Let MM be the midpoint of BCBC. A variable point PP is selected on segment AMAM. The circumcircles of triangles BPMBPM and CPMCPM meet Ω\Omega again at points DD and EE, respectively. The lines DPDP and EPEP meet the circumcircles of triangles CPMCPM and BPMBPM again at points XX and YY, respectively. Prove that, as PP varies, the circumcircle of triangle AXYAXY passes through a fixed point TT distinct from AA.
Step 5 of 9: The chord XY is parallel to DE
Previous stepNext step
∠LXY=∠LBC=∠BCN=∠NDE ⟹ XY∥DE\angle LXY=\angle LBC=\angle BCN=\angle NDE\ \Longrightarrow\ XY\parallel DE
Detailed analysis

Because B,Y,X,CB,Y,X,C are cyclic (step 4), and L,B,YL,B,Y are collinear, ∠LXY=∠LBC\angle LXY=\angle LBC. Also LB∥CELB\parallel CE (step 2), while C,N,EC,N,E are collinear by the radical-center construction, so ∠LBC=∠BCN\angle LBC=\angle BCN. Finally B,C,D,EB,C,D,E lie on Ω\Omega, and B,N,DB,N,D are collinear, hence ∠BCN=∠BCE=∠BDE=∠NDE\angle BCN=\angle BCE=\angle BDE=\angle NDE. Therefore ∠LXY=∠NDE\angle LXY=\angle NDE, which gives XY∥DEXY\parallel DE in directed angles.