Trinity Force and Infinity Edge build paths.

There are numerous ways that Champions can scale into lategame as a main damage source for a team. The two most common routes are through Ability Power (APCs) and Attack Damage (ADCs). These two categories are very generalized and each Champion has different abilities that scale better with certain items. No two Champions have the exact same build path. On the AD oriented side, there are Champions that scale better with Infinity Edge and those that scale better with Trinity Force. (Some of the things I will be saying will be generalized because of the number of outlier Champions such as Garen who scales with Crit, but not AAs.)

Infinity Edge Champions (IECs) require 6 stats in order to deal massive Damage with their Auto Attacks. One cannot remove a single stat without losing maximum efficiency in terms of damage. These 6 stats are Attack Damage, Attack Speed, Lifesteal, Critical Strikes, Critical Damage, and Armor Penetration (and movementspeed to a lesser extent). Ranged Attack Damage Carries (RADCs) are the Champions that use the Infinity Edge build path the most. Their early game is weak because they don't have impressive base stats or huge steroids outside of attack speed. But their late game is very strong once they have a full build. Their Late Game DPS can reach anywhere from around 600 to 1200 physical DPS. We can call this Quadratic Scaling because of how their Damage Output curves upwards drastically with each item they buy. The downside to IECs is that they are very squishy and prone to being burst 100% - 0% Health by APCs or being beatdown by Offtanks and Tanks. A dead RADC is a useless RADC.

Trinity Force Champions (TFCs) require only a Trinity Force and a low cooldown ability. TFCs are strange Champions because they do not require large amounts of AD or AS or Lifesteal or Critical Strikes or Critical Damage. The reason that TFCs do not need to buy these items is because they are using a different sort of scaling. The Sheen Proc does cause a spike in damage every two seconds. The damage scaling on Trinity Force Scales with the Champion's Base AD which is tied to levels. Thus, we can consider this to be Linear Scaling. TFCs have a strong Early-Mid Game to Mid-Late Game because their damage spikes early and plateaus, increasing very slightly afterwards. Most TFCs are melees and scale better into Late Game with Cooldown Reduction (CDR) and Health and Resistances and some Lifesteal. With only a single Trinity Force as a damage item, it gives Champions that can make use of the Sheen Proc near equivalent Damage to an ADC.

Mathematics of the Ideal Average Damage: Trinity Force versus Infinity Edge
(Damage is calculated over 100 Auto Attacks)

Trinity Force: Total Damage over 100 Auto Attacks: Assuming 100 Base AD + 30 AD + 10% Crit from TF. Sheen Proc = 2 X Base AD.
A)130 AD x 90 =11,700 Non-Crit AA
B)130 AD X 2 (Crit Damage) X 10 (10% Crit) = 2,600 AA Crit Damage
C)Sheen = 2 X Base AD= 2 X 100 AD = 200 Sheen Damage every 2 seconds. 200 X 50 Procs (1 Proc per 2 Autos) = 10,000
-A)11,700 + B)2,00 + C)10,000 =24,300 Physical Damage from Trinity Force
Average: 243 Damage per Auto Attack

Infinity Edge: Totla Damage over 100 Auto Attacks: Assuming 100 Base AD + 80 AD + 25% Crit from IE.
A)180 AD X 75 = 13,500
B)180 AD X 2.5 (Crit Damage) X 25 (25% Crit) = 11,250 Crit Damage
-A)13,500 + B)11,250 = 24,750 Physical Damage from IE
Average: 247.5 Damage per Auto Attack

As we can see, the Damage is near equal if a Champion were to buy one item or the other. Trinity Force's Sheen Proc allows the TFC to get approximately equivalent damage to an IEC. However, I have not even listed the numerous other stats such as AP, AS, Health, Mana, Flat and % Movementspeed that Trinity Force gives. Now by my line of reasoning, Trinity Force seems to be the better of the two, but the IEC's DPS increases exponentially as the IEC gets closer to finished. In the Late Late Game, Infinity Edge Champions will outscale Trinity Force Champions in terms of Damage by a slight amount.

P.S.: On a side note, Trinity Force actually scales better with CDR because it increases the chance that the TFC will proc Sheen every time. If Riot decides to add more items that have both CDR and AD and/or AP, Trinity Force will scale even better into the Lategame. Tons of Damage. =)