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National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
1999 All-Russian Olympiad Regional Round
9.8
9.8
Part of
1999 All-Russian Olympiad Regional Round
Problems
(1)
QO_|_AC if QP _|_ KM, QM//BO - All-Russian MO 1999 Regional (R4) 9.8
Source:
9/25/2024
In triangle
A
B
C
ABC
A
BC
(
A
B
>
B
C
AB > BC
A
B
>
BC
),
K
K
K
and
M
M
M
are the midpoints of sides
A
B
AB
A
B
and
A
C
AC
A
C
,
O
O
O
is the point of intersection of the angle bisectors. Let
P
P
P
be the intersection point of lines
K
M
KM
K
M
and
C
O
CO
CO
, and the point
Q
Q
Q
is such that
Q
P
⊥
K
M
QP \perp KM
QP
⊥
K
M
and
Q
M
∥
B
O
QM \parallel BO
QM
∥
BO
. Prove that
Q
O
⊥
A
C
QO \perp AC
QO
⊥
A
C
.
geometry
perpendicular