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Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2017 South Africa National Olympiad
5
5
Part of
2017 South Africa National Olympiad
Problems
(1)
Intersection lies on circumcircle
Source: SAMO 2017 Q5
7/30/2017
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
Γ
\Gamma
Γ
. Let
D
D
D
be a point on segment
B
C
BC
BC
such that
∠
B
A
D
=
∠
D
A
C
\angle BAD = \angle DAC
∠
B
A
D
=
∠
D
A
C
, and let
M
M
M
and
N
N
N
be points on segments
B
D
BD
B
D
and
C
D
CD
C
D
, respectively, such that
∠
M
A
D
=
∠
D
A
N
\angle MAD = \angle DAN
∠
M
A
D
=
∠
D
A
N
. Let
S
,
P
S, P
S
,
P
and
Q
Q
Q
(all different from
A
A
A
) be the intersections of the rays
A
D
AD
A
D
,
A
M
AM
A
M
and
A
N
AN
A
N
with
Γ
\Gamma
Γ
, respectively.Show that the intersection of
S
M
SM
SM
and
Q
D
QD
Q
D
lies on
Γ
\Gamma
Γ
.
geometry
circumcircle