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National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1962 Bulgaria National Olympiad
Problem 4
Problem 4
Part of
1962 Bulgaria National Olympiad
Problems
(1)
point in triangle, distance to sides and vertices (Bulgaria 1962 P4)
Source:
6/24/2021
There are given a triangle and some internal point
P
P
P
.
x
,
y
,
z
x,y,z
x
,
y
,
z
are distances from
P
P
P
to the vertices
A
,
B
A,B
A
,
B
and
C
C
C
.
p
,
q
,
r
p,q,r
p
,
q
,
r
are distances from
P
P
P
to the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Prove that:
x
y
z
≥
(
q
+
r
)
(
r
+
p
)
(
p
+
q
)
.
xyz\ge(q+r)(r+p)(p+q).
x
yz
≥
(
q
+
r
)
(
r
+
p
)
(
p
+
q
)
.
geometry
Triangles