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Mathematics
Problem
The determinant
∣
a
b
a
c
+
b
b
c
b
c
+
c
a
c
+
b
b
c
+
c
0
∣
\begin{vmatrix} a & b & ac + b \\ b & c & bc + c \\ ac + b & bc + c & 0 \end{vmatrix}
a
b
a
c
+
b
b
c
b
c
+
c
a
c
+
b
b
c
+
c
0
is equal to zero, if a.
a
,
b
a, b
a
,
b
and
c
c
c
are in AP b.
a
,
b
,
c
a, b, c
a
,
b
,
c
are in GP c.
a
,
b
a, b
a
,
b
and
c
c
c
are in HP d.
(
a
−
c
)
(a-c)
(
a
−
c
)
is a factor of
a
c
2
+
2
b
c
+
c
ac^2 + 2bc + c
a
c
2
+
2
b
c
+
c
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