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Turkey Contests
Turkey Team Selection Test
2023 Turkey Team Selection Test
9
9
Part of
2023 Turkey Team Selection Test
Problems
(1)
Geometry with Spiral Similarity from Turkey TST
Source: 2023 Turkey D3 P9
3/29/2023
The points
A
,
B
,
K
,
L
,
X
A,B,K,L,X
A
,
B
,
K
,
L
,
X
lies of the circle
Γ
\Gamma
Γ
in that order such that the arcs
B
K
^
\widehat{BK}
B
K
and
K
L
^
\widehat{KL}
K
L
are equal. The circle that passes through
A
A
A
and tangent to
B
K
BK
B
K
at
B
B
B
intersects the line segment
K
X
KX
K
X
at
P
P
P
and
Q
Q
Q
. The circle that passes through
A
A
A
and tangent to
B
L
BL
B
L
at
B
B
B
intersect the line segment
B
X
BX
BX
for the second time at
T
T
T
. Prove that
∠
P
T
B
=
∠
X
T
Q
\angle{PTB} = \angle{XTQ}
∠
PTB
=
∠
XTQ
geometry