Solution:
It is given that
b is the mean proportional between a and c
We can write it as
b2 = a × c
b2 = ac ….. (1)
We know that
a, c, \[{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}~\mathbf{and}\text{ }{{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}~\]are in proportion
It can be written as
\[a/c\text{ }=\text{ }({{a}^{2}}~+\text{ }{{b}^{2}})/\text{ }({{b}^{2}}~+\text{ }{{c}^{2}})\]
By cross multiplication
\[a\text{ }({{b}^{2}}~+\text{ }{{c}^{2}})\text{ }=\text{ }c\text{ }({{a}^{2}}~+\text{ }{{b}^{2}})\]
Using equation (1)
\[a\text{ }(ac\text{ }+\text{ }{{c}^{2}})\text{ }=\text{ }c\text{ }({{a}^{2}}~+\text{ }ac)\]
So we get
\[ac\text{ }\left( a\text{ }+\text{ }c \right)\text{ }=\text{ }{{a}^{2}}c\text{ }+\text{ }a{{c}^{2}}\]
Here ac (a + c) = ac (a + c) which is true.
Therefore, it is proved.