Achilles is a name, so you can use it the same way as any other name.
Society's achilles' heel is the politician. (Since heel is singular, so is "is", and not are.)
The phrase "Achilles' heel" is a way of naming a weakness. An example of a sentence using the phrase would be: While he was a fantastic Basketball player, free throws were his Achilles' heel.
His inability to endure long distance running is his Achilles heel.
When it comes to school subjects, math is definitely my Achilles heel.
In the human being, the Achilles tendon attaches the calf muscles to the heel bone.I am stretching my Achilles tendon.
It is said that Achilles used yarrow to treat the wounds of his soldiers after the battle of Troy.
The palm reader made a prophecy that the man would become rich and have a good life.
If you are indicating possession (Achilles' heal) use an apostrophe at the end of the word. If you are simply stating his name, there is no apostrophe.
Achilles
Her bravery on the battlefield evoked images of ancient warriors like Achilles or Joan of Arc.
Odysseus did not kill Achilles. Paris killed Achilles.
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