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Romania TST 2022 Day 2 P2

Source: Romania TST 2022

May 15, 2022

geometryromaniaRomanian TST

Problem Statement

Let

ABC

be an acute triangle and let

B'

and

C'

be the feet of the heights

B

and

C

of triangle

ABC

respectively. Let

B_A'

and

B_C'

be reflections of

B'

with respect to the lines

BC

and

AB

, respectively. The circle

BB_A'B_C'

, centered in

O_B

, intersects the line

AB

in

X_B

for the second time.
The points

C_A', C_B', O_C, X_C

are defined analogously, by replacing the pair

(B, B')

with the pair

(C, C')

. Show that

O_BX_B

and

O_CX_C

are parallel.

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