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Problem
Problem
Let
P
P
P
be the point inside
Δ
A
B
C
\Delta ABC
Δ
A
B
C
such that
∠
A
P
B
=
∠
B
P
C
=
∠
C
P
A
\angle APB = \angle BPC = \angle CPA
∠
A
P
B
=
∠
B
P
C
=
∠
C
P
A
. Prove that
P
A
+
P
B
+
P
C
=
a
2
+
b
2
+
c
2
2
+
2
3
Δ
,
PA + PB + PC = \sqrt{\frac{a^2 + b^2 + c^2}{2} + 2\sqrt{3}\Delta},
P
A
+
P
B
+
P
C
=
2
a
2
+
b
2
+
c
2
+
2
3
Δ
,
where
a
,
b
,
c
,
Δ
a, b, c, \Delta
a
,
b
,
c
,
Δ
are the sides and the area of
Δ
A
B
C
\Delta ABC
Δ
A
B
C
.
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