Talk:2021 USAMO Problems/Problem 1
We are given the acute triangle AA_1B_2B, BB_1C_2C, CC_1A_2A
\angle AA_1B + \angle BB_1C + \angle CC_1A = \pi
\angle AA_1B = \alpha, \angle BB_1C = \beta, \angle CC_1A = \gamma$.
Construct circumcircles$ (Error compiling LaTeX. Unknown error_msg)X_1, X_2AA_1B_2B, BB_1C_2C
X_1, X_2
B
O
A_1B
X_1
C_2B
X_2
\angle A_1OB = \angle C_2OB = \frac{\pi}{2}
\angle A_1OC_2 = \pi
O
A_1C_2
\angle A_1BB_2 = \angle A_1OB_2 = \alpha
X1
\angle B_1BC_2 = \angle B_1OC_2 = \beta
\angle B_2OB_1 = \pi - (\alpha + \beta) = \gamma$.
Construct another circumcircle around the triangle$ (Error compiling LaTeX. Unknown error_msg)AOCAA_2
A'
CC_1
C'
A'=A_2, C'=C_2
\angle A2CC_1 = \gamma
\angle A'OC' = \angle A'AC' = \gamma
A'
AA_2
\angle A_2AC' = \angle A_2CC_1
C'
AC_1
C'
CC_1
C'=C1
A'=A_2$.
Finally, since$ (Error compiling LaTeX. Unknown error_msg)\angle A_2AC = \frac{\pi}{2}A_2C
X_3
\angle A_2OC = \frac{\pi}{2}
\angle B_1OC = \angle C_1OA = \angle B_2OA = \frac{\pi}{2}
B_2C_1, B_1A_2$, and the three diagonals are concurrent.