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National and Regional Contests
Turkey Contests
Turkey EGMO TST
2019 Turkey EGMO TST
5
5
Part of
2019 Turkey EGMO TST
Problems
(1)
Turkey EGMO TST 2019 P5
Source: Turkey EGMO TST 2019
3/15/2019
Let
D
D
D
be the midpoint of
B
C
‾
\overline{BC}
BC
in
Δ
A
B
C
\Delta ABC
Δ
A
BC
. Let
P
P
P
be any point on
A
D
‾
\overline{AD}
A
D
. If the internal angle bisector of
∠
A
B
P
\angle ABP
∠
A
BP
and
∠
A
C
P
\angle ACP
∠
A
CP
intersect at
Q
Q
Q
. Prove that, if
B
Q
⊥
Q
C
BQ \perp QC
BQ
⊥
QC
, then
Q
Q
Q
lies on
A
D
AD
A
D
geometry