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Hungary Contests
Kürschák Math Competition
2015 Kurschak Competition
2
2
Part of
2015 Kurschak Competition
Problems
(1)
Lines intersecting circles
Source: Kürschák 2015, problem 2
10/7/2016
Consider a triangle
A
B
C
ABC
A
BC
and a point
D
D
D
on its side
A
B
‾
\overline{AB}
A
B
. Let
I
I
I
be a point inside
△
A
B
C
\triangle ABC
△
A
BC
on the angle bisector of
A
C
B
ACB
A
CB
. The second intersections of lines
A
I
AI
A
I
and
C
I
CI
C
I
with circle
A
C
D
ACD
A
C
D
are
P
P
P
and
Q
Q
Q
, respectively. Similarly, the second intersection of lines
B
I
BI
B
I
and
C
I
CI
C
I
with circle
B
C
D
BCD
BC
D
are
R
R
R
and
S
S
S
, respectively. Show that if
P
≠
Q
P\neq Q
P
=
Q
and
R
≠
S
R\neq S
R
=
S
, then lines
A
B
AB
A
B
,
P
Q
PQ
PQ
and
R
S
RS
RS
pass through a point or are parallel.
geometry
angle bisector