Yes they do.
Go to 'view all sizes'.
Click on 'medium 500'.
Click on the image--depending on your web browser, your action needs to be a little different--for me, in firefox on the mac, I click and hold over the image to see a dropdown menu that includes 'copy image location'; for a windows machine, you might need to right-click in internet explorer. But you want the actual image location, the one that has 'farmx.static.flickr.com and lots of letters and numbers.
Back in tea chat, click the 'IMG' button. Paste your image location between them.
If you want to be advanced about it, highlight the entire thing you just entered in the reply box--from [IMG....... to ........IMG] and then click 'URL' button. Now, in the first URL bracket, put in an equals sign right after URL=, and paste in the link to the flickr page featuring the photo. Now you're created a link to your image that clicks and takes you back to the flickr image page.
Want to see what it actually looks like? Just hit 'quote' for this post and see what my link looks like.