What is a tangent line and when was it first defined?
In Apollonius' work on Conics (c. 225 BC) a tangent line is defined as a line such that no other straight line could fall between it and the curve.
The fools of mainstream mathematics academia are not even worthy to lick Apollonius' rear end. In fact, the majority have never even completed a study of his historic work on Conics. Those who tried, have done so without much success. Besides Sir Thomas Heath, there haven't really been any others.
Using that definition, it would not even be relevant to talk about a line crossing at the point of tangency. The Ancient Greeks called the tangent line ἐφαπτομένη. In the Greek, the word has similar meaning to νεῦσις which means "verging toward" or "inclining toward". The word's meaning was so well known and commonly understood that Euclid didn't bother defining it. The original meaning is that there cannot be another line between the tangent and a non-linear curve. Tangents go hand in hand with conic curves. A chief property of a tangent line is in determining the smoothness of a given curve over an interval.
The Ancient Greeks defined the tangent line correctly (without any reference to derivatives) and the mainstream math monkeys of the last 150 years imagine they know better.
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