Middle English tēchen, from Old English tæcan (past tense tæhte, past participle tæht) "to show (transitive), point out, declare; demonstrate," also "give instruction, train, assign, direct; warn; persuade."
This is reconstructed to be from Proto-Germanic *taikijan "to show" (source also of Old High German zihan, German zeihen "to accuse," Gothic ga-teihan "to announce"), from PIE root *deik- "to show, point out." It is related to Old English tacen, tacn "sign, mark" (see token). The notion is "to show how to do something by way of information or instruction." Related: Taught; teaching.
By mid-14c. as "disseminate" a system of belief. By c. 1200 as "indicate" how something is to happen; used by 1560s in threats, "make known to one at a cost."
Enraged lemonade vendor (Edgar Kennedy): I'll teach you to kick me!
Chico: you don't have to teach me, I know how. [kicks him]
["Duck Soup," 1933]
The usual sense of Old English tæcan was "show, declare, warn, persuade" (compare German zeigen "to show," from the same root); while the Old English word for "to teach, instruct, guide" was more commonly læran, source of modern learn and lore.
