I like to do dips(for triceps and back), bench press (for triceps and pecs) and overhead press (for triceps and delts) when I do my upper body workout, generally three times / week. However after doing any of the three to fatigue, my triceps are my limiting factor that don't allow me to do the other two exercises. I almost fell off the dip rack the other day when I tried to do dips after bench pressing and overhead presses and my triceps gave out.

Should I just do one of the three exercises during a workout and then rest for two days before doing the second exercise? If so, then I can only realistically do one of each exercise / week.

Or, should I back off on the intensity of each and try to work all three in during a workout?

Or is there a better suggestion?

Your triceps need to get stronger and in better shape. End each workout with a high volume triceps exercise. For about a month. Press-Down for time, overhead extensions, dumbbell extensions, it does not really matter the exercise.

And you might want to pick one low volume movement to specialize in and one movement to work move volume.

What is your training experience and what are your goals?

But, after doing the dips, bench press, or overhead press, my triceps are shot. I don't think I can do those tricep isolation exercises.

What is a low volume movement? I don't understand these terms.

I have been working out for about 1 year. I started with a personal trainer last August and worked with him for about 3 months, then I had an angioplasty in November, took off 6 weeks and started again at the beginning of the year. I try to learn from others as much as I can and feel I am reasonably knowledgeable now.

As for goals, they are to look good and to be able to perform well in cycling and, eventually, short (sprint) triathlons.

You can do them. It does not have to be really heavy, you do need the work though. 5 - 8 minutes at the end of your workout.

Low volume is another way of saying lower reps. (when referencing an individual exercise) So sets of 3 would be considered low volume. Sets of 15 is medium volume. More than 20 is higher volume.

Volume when referencing a workout would be the total amount of sets an reps. So a workout using all 3 - 5 reps for 3 sets per exercise would be considered a low volume workout. Using 3 - 5 reps for 8 sets per exercise would be more volume. Using 15 reps and 4 sets per exercise would be a pretty high volume.

I typically stick with one high intensity (3-5 sets of 3-8 reps) compound pushing movement per workout session and then 2-4 of sets of 8-15 reps of a triceps isolation exercise at the end. I alternate a horizontal pushin movement one session with a vertical pushing movement on alternate sessions. For the triceps "volume" work I might do 2 sets of lying triceps extensions and 2 sets of reverse pushups to end a session.