So I'm now of the belief that Penguin's strange triangular scar was the work of the Joker. I think Joker was cutting the skin from his mouth to leave him with a gnarly full toothed grin, but was interrupted during the process and Penguin was saved from an agonizing death. It's possible Joker also shot him in the leg and gave him his trademark waddle that requires a leg brace. We know how Joker likes to cripple people...
On the scar: This is clearly multiple cuts, and not slashes you'd get in a brawl, w/ the top part going from his upper lip like a cleft lip and outwards to his cheek being the deepest of the cuts. You can see the weird bits in the corner which is the result of suturing a deep cut (you see this on Ledger's glasgow smile in TDK)/
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And here's my terrible mockup of what this could have looked like if completed.
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And here is one of Joker's victims from the comics: