2021 USAMO Problems/Problem 1
Rectangles 
and 
are erected outside an acute triangle 
Suppose that![\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]](http://latex.artofproblemsolving.com/3/3/7/3378f0c4409d14bc0afddf2616992c13f624be3b.png)
Prove that lines 
and 
are concurrent.
Solution
Let 
be the second point of intersection of the circles 
and 
Then
![\[\angle ADB = 180^\circ – \angle AB_1B,\angle ADC = 180^\circ – \angle AA_1C \implies\]](http://latex.artofproblemsolving.com/3/0/e/30ec491537c11dc09077e026685fe4d7a8e0a92d.png)
![\[\angle BDC = 360^\circ – \angle ADB – \angle ADC = 360^\circ – (180^\circ – \angle AB_1B) – (180^\circ – \angle AA_1C) =\]](http://latex.artofproblemsolving.com/1/d/6/1d62306a8e82a2e2ef9b4189e59954d2fca94f7b.png)
![\[=\angle AB_1B + \angle AA_1C \implies \angle BDC + \angle BC_1C = 180^\circ \implies\]](http://latex.artofproblemsolving.com/9/e/5/9e5c0e75f31c0ba4583a17b89a26114bc0550849.png)
is cyclic with diameters 
and 
