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Problems
Contests
National and Regional Contests
Israel Contests
Israel Olympic Revenge
2021 Israel Olympic Revenge
3
3
Part of
2021 Israel Olympic Revenge
Problems
(1)
Two lines intersent on a circle
Source: Israeli Olympic Revenge 2021, Problem 3
8/29/2021
Let
A
B
C
ABC
A
BC
be a triangle. A point
P
P
P
is chosen inside
△
A
B
C
\triangle ABC
△
A
BC
such that
∠
B
P
C
+
∠
B
A
C
=
18
0
∘
\angle BPC+\angle BAC=180^{\circ}
∠
BPC
+
∠
B
A
C
=
18
0
∘
. The lines
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
intersect
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
P
A
,
P
B
,
P
C
P_A,P_B,P_C
P
A
,
P
B
,
P
C
respectively. Let
X
A
X_A
X
A
be the second intersection of the circumcircles of
△
A
B
C
\triangle ABC
△
A
BC
and
△
A
P
B
P
C
\triangle AP_BP_C
△
A
P
B
P
C
. Similarly define
X
B
,
X
C
X_B,X_C
X
B
,
X
C
. Let
B
′
B'
B
′
be the intersection of lines
A
X
A
,
C
X
C
AX_A,CX_C
A
X
A
,
C
X
C
, and let
C
′
C'
C
′
be the intersection of lines
A
X
A
,
B
X
B
AX_A,BX_B
A
X
A
,
B
X
B
. Prove that lines
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
intersect on the circumcircle of
△
A
P
B
P
C
\triangle AP_BP_C
△
A
P
B
P
C
.
geometry
circumcircle