Hexagonal Triangles!

Charcoal, 62 43 39 bytes

ＮβＦ³«Ｆ⁻β‹ι²« / \¶|   |¶ \ /Ｍײι↑¿⁻¹ιＭ⁶←

Try it online! Edit: Saved some bytes thanks to @ASCII-only (not sure exactly how many as I also saved another 6 10 bytes but then needed to spend 4 bytes on a bugfix). Now using nested loops, which is what this should have been all along.

Python 2, 184 177 174 164 bytes

n=input();p='|   |';e=' \\ /'
for i in range(n):k=' '*(4*i-5);w='  '*~(i-n);print w+[' / \\'*-~i+'\n'+w+p[:4]*i+p,' /'+e+k[3:]+e+' \\\n'+w+p+k+p][2<i<n-1]
print e*n

Try it Online!

-7 bytes thanks to Ørjan Johansen

JavaScript (ES6), 243 bytes

f=n=>(n=n*2-2,a=[...Array(n+3)].map(_=>Array(n*2+5).fill` `),g=(y,x)=>(a[y+1][x]=a[y+1][x+4]=`|`,a[y][x+1]=a[y+2][x+3]=`/`,a[y][x+3]=a[y+2][x+1]=`\\`),[...Array(n+!n)].map((_,i)=>i%2||(g(n,i*2),g(i,i=n-i),g(i,n+i))),a.map(a=>a.join``).join`
`)

<input type=number oninput=o.textContent=f(this.value)><pre id=o>