Leuenberger's inequality

Question

Leuenberger's inequality states that the sum of the medians of a triangle is at most the sum of its exradii. I understand how to prove this when the triangle is acute. However, the proof I've seen in Bottema et. al., or even here: https://www.cut-the-knot.org/triangle/LeuenbergerInequality.shtml is wrong (look at it carefully, it doesn't work for an obtuse triangle since one of the cosines is negative, so $p_a+p_b+p_c$ is greater than $R+r$ in this case. In fact, if $C$ is the obtuse angle then $p_a+p_b-p_c=R+r$). How to fix the proof for obtuse triangles? and, is there a simple unified proof for all triangles?