Where does homebrew install the android-sdk when you run the command:
brew cask install android-sdk
I want to be able to add it to the PATH
Unfortunately, brew has now changed where android-sdk is installed, so all of these older questions are no longer correct.
When installing via brew cask install android-sdk, you'll now be informed that you should add export ANDROID_SDK_ROOT=/usr/local/share/android-sdk to your profile, and, indeed, that directory is now a symlink to the version android-sdk directory (currently /usr/local/Caskroom/android-sdk/25.2.3).
To summarise, add the following to your .profile/.bashrc/.whatever:
export ANDROID_SDK_ROOT=/usr/local/share/android-sdk
export PATH=$PATH:$ANDROID_SDK_ROOT/tools:$ANDROID_SDK_ROOT/platform-tools