So is the HDMI and displayport both on your 7870 video card?
How specifically is the displayport connection between the monitor and the computer - meaning, do you use an adapter, and what kind? If the adapter is active or passive, that will make a difference.
Let's assume it's passive. Then you can connect the 3rd display to the video-out of your motherboard, instead of the video card. So you'll have the existing dual-display on the 7870, and add a 3rd monitor using the motherboard video out. Have you activated the motherboard's video output?
Let's assume your displayport adapter is active. Then you can connect the 3rd display anywhere it would fit, either on the 7870 or the motherboard.