The word tangent names two different things: a length measured on the circle and the line that touches a curve. They are not synonyms, and they are not mere homonyms either — they meet exactly once, in a single identity, and that meeting is what this sheet illustrates.
Do: walk with x and look at the angle θ: what the tangent line does with the floor is the same angle there in the circle. The golden segment on the left has the same length as the golden leg on the right — and since the two panels divide the scale, that checks out with the ruler. the names of the triangle shows who is who.
1. The two gold segments are always the same length. On the left is the tangent segment of the circle with radius 1 — the tan θ that the circle slice measured. On the right is how much the curve rises when you move 1 to the right along the tangent line. They are the same number, and that's why f ′(x) = tan θ.
2. It's the triangle of always. Connect the names of the triangle: the adjacent leg is 1 and the opposite is f ′(x). And tangent is opposite over adjacent — but since the adjacent is 1, the division does nothing, and the tangent is the opposite. The same triangle from o-triangulo.html and the same OAT from o-circulo.html.
3. Where the curve is flat, the circle point is on the ground. At a maximum or a minimum, the tangent line becomes horizontal: θ = 0, the gold segment disappears, and f ′ = 0. Where the curve descends, θ is negative and the segment points downward. The slope is not a metaphor — it is an angle, and it is drawn.
A color changes here, on purpose. In the other slices the leg that rises is terracotta — it's the perpendicular component. Here it is golden, the color of the tangent, because the thesis of the slice is exactly that it is the segment of tangent of the circle. The change is the argument, and so it is declared instead of happening silently.
The tangent line got its name because it touches; the length on the circle got its name because it is measured along the line that touches the circle. They come from the same Latin word and name different objects — but they are not a mere coincidence of vocabulary: they come together in this equality, and nowhere else. It is the "Where the Two Meet" sheet of C1, and what the entire arc of calculus uses from C2.
Local instrument, self-contained: no network, no external library. It's where the two meanings
of the word tangent meet — the length in the circle, which the slices of measure built,
and the line that touches the curve, which the rate showed walking.