Orthogonality Preserving Linear Map.

I'm having trouble with the following problem:

Let $V$ be a real vector space equipped with an inner product $\langle\cdot,\cdot\rangle$ (and thus with a norm induced by that inner product).
Let $T:V\to V$ be a bijective linear transformation such that if $\langle u,v\rangle=0$ , then $\langle T(u),T(v)\rangle=0$ .

Given an orthonormal basis $\{b_1,...,b_n\}$ is $V$ , it clearly is the case that $\{T(b_1),...,T(b_n)\}$ is an orthogonal basis of $V$ and hence $\left\{\frac{T(b_1)}{\|T(b_1)\|},...,\frac{T(b_n)} {\|T(b_n)\|}\right\}$ is an orthonormal basis.

I now have to show that $\|T(b_1)\|=\cdots=\|T(b_n)\|$ , but everything I've tried so far has lead me to a dead end.

Any hint or tip from someone who knows how to prove this would be greatly appreciated (I'm not looking for someone to post a complete solution).