How are Tai Chi Tae Kwon Do and Aikido related?
Historically there is some connection between all of them having shared roots, yet being developed separately in each country of their origin.
Taijiquan (T'ai Chi Ch'uan) would be the older of these martials art systems by name, but all Martial Art curricula were influenced by earlier, undocumented sources and have each evolved over time. Taijiquan is believed to have evolved out of the systems practiced among five Chinese families.
The Chinese Martial Art that developed in the Shaolin Temples were also a blend of earlier sourcdes. Legend has it that the a Buddhist monk brought the basic principles of the philosophy and some hand strike techniques for physical exercise training from India and taught it to monks in the Shoalin Temple in China, however there is also evidence that fighting skills were prevalent in China for centuries prior. China's hand fighting influenced both the Korean and Japanese systems.
Aikido and Taekwondo are more recent creations in their modern form of instruction, though both are based on centuries ancient principles and training that existed in each of their countries respectively.
Aikido is a Japanese art of evasion and redirecting while blending with an opponent's forces with some grappling controls utilized. It evolved out of earlier Japanese systems that can also be traced back to influences from China and even Korea since all of these Asian cultures shared information regularly throughout history.
Taekwondo is primarily a striking art with a blend of throwing and grappling skills. It employs hand and foot strikes similar to Japanese karate, but places the priority of techniques with long range first (kicks then hands - a philosophy of traditional Korean kicking called Tae Kkyeon), followed by close range (Knees and elbows), then joint manipulations, throws, and finally ground-fighting. The techniques of close contact grappling in Taekwondo are partially influenced by older Korean arts, such as subak, but with more recent influences of Japanese Judo (called Yudo in Korea), and Aikido (Hapkido). Some of the influences that led to the development of Taekwondo's first school (Chung Do Kwan) came from the Chinese Hand fighting (Tangsudo), which was based on training going back to the Shoalin Temples and shared origins with T'ai Chi Ch'uan.
Perhaps the main relationship connection between all of these Asian Martial Art systems can be attributed to a shared philosophy and teachings of the I Ching - Book of Changes. The principles of the Chinese T'ai Chi Tu (Yin and Yang circle), and Korean Taegeuk (Um and Yang) all come from the same source and are core philosophies of these Martial Art systems.