  A matroid $N$ is said to be triangle-rounded in a class of matroids
$\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle
$T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result
useful to identify such matroids as stated next: suppose that $M$ is a binary
$3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$
and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an
$N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. We
strengthen this result by dropping the condition that such element $e$ exists
and proving that there is a $3$-connected minor $M'$ of $M$ with an $N$-minor
$N'$ such that $T$ is a triangle of $M'$ and $E(M')-E(N')\subseteq T$. This
result is extended to the non-binary case and, as an application, we prove that
$M(K_5)$ is triangle-rounded in the class of the regular matroids.
