Many bloggers and marketers recommend republishing your blog on Medium because it gets your content in front of new audiences and apparently that works really well.
But this poses a huge SEO problem - duplicate content. Search engines might think the original content is the one being posted on Medium. This might hurt your website rankings. Generally, there's a workaround - the "rel=canonical" tag that basically tells Google which URL of the two is "original" - but you cannot use this tag when posting to Medium directly via the web interface. But you can add this tag when posting via the Medium API or via IFTTT (which is basically another way of using the API).
What you'll need:
https://api.medium.com/v1/me?accessToken=XXXX where "XXXX" is the access token from above, and find your user ID in the response JSON.
Then use whichever is your favorite way of accessing REST APIs. Here's a short Node.js script I wrote to post stuff to Medium: https://github.com/jitbit/MediumNodeJs/blob/master/Post.js
Note the “canonicalUrl” parameter
Another way to republish your blog content on Medium is via ifttt.com which also supports “canonicalUrl”.
The above "recipe" will only publish new posts to Medium, so if you want to publish some of your existing content you still need to use the API manually.