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National and Regional Contests
Poland Contests
Poland - Second Round
1988 Poland - Second Round
3
3
Part of
1988 Poland - Second Round
Problems
(1)
min |BL|^2 + |CM|^2 + |AN|^2
Source: Polish MO Recond Round 1988 p3
9/9/2024
Inside the acute-angled triangle
A
B
C
ABC
A
BC
we consider the point
P
P
P
and its projections
L
,
M
,
N
L, M, N
L
,
M
,
N
to the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Determine the point
P
P
P
for which the sum
∣
B
L
∣
2
+
∣
C
M
∣
2
+
∣
A
N
∣
2
|BL|^2 + |CM|^2 + |AN|^2
∣
B
L
∣
2
+
∣
CM
∣
2
+
∣
A
N
∣
2
is the smallest.
geometry
Geometric Inequalities