Arduino as on/off button for external circuit

I have a projector mounted to the ceiling, the power button is one click on, one click off and I'd like to control this using an arduino. I don't think I'd be able to just use a relay because even if the power of the projector is on the projector isn't necessarily on. The ultimate goal is to use the amazon alexa scripts to make the arduino simulate one button push for on and one for off but how do I do this without the arduino integrated to the circuit? I'm sure there is a chip out there to solve my issue, but my have so little knowledge on the subject that I'm not even sure what to search for..
Found a video that might suggest I use the CD4066 IC, but this seems more advance than I would need. Thanks for the help in advance!

You'll want to look at an Arduino controlling a solenoid which would simulate a finger pushing the button.

Are you talking about physically pushing the button on the projector on the ceiling or pushing the on/off button on a remote that talks to the projector on the ceiling?

@blh64

blh64:
Are you talking about physically pushing the button on the projector on the ceiling or pushing the on/off button on a remote that talks to the projector on the ceiling?

I suppose it might be easier to just transmit the matching IR signal at it, but I was really thinking about removing the button and inserting an arduino circuit without physically pushing the button with a servo or something.

Internet control of a button...
I would probably use a small WiFi enabled 'Arduino', like the WeMos D1 mini, or NodeMCU etc., powered by a 5volt cellphone charger. And let that drive an opto coupler, with it's transistor connected across the on/off button of the device.
Local control (PC/laptop/smartphone) and remote internet control (Alexa?) should be possible (no experience in the latter).
Leo..