Guidelines to reduce general TSP to Triangle TSP
Here is a simple reduction for the TSP problem to the metric TSP problem:

For the given TSP instance with $n$ cities, let $D(i,j) \geq 0$ denote the distance between $i$ and $j$. Now let $M = \max_{i,j} D(i,j)$. Define the metric TSP instance by the distances $D'(i,j) := D(i,j)+M$. To see that this gives a metric TSP instance, let $i,j,k$ be arbitrary. Then $D'(i,j) + D'(j,k) = D(i,j) + D(j,k) + 2M \geq 2M \geq D(i,k) + M = D'(i,k)$. Since any tour uses exactly $n$ edges, the transformation adds exactly $nM$ to any tour, which shows the correctness of the reduction.

Remark: We can of course also allow for negative distances in the original TSP instance if you prefer by changing the reduction slightly.