**Guinea-Bissau 1979** Michel 529A Scott 412

**Inscribed Angle Theorem**

Mathematics on the blackboard: A fine and quite rare theme on stamps. The teacher explains classical theorems in Euclidean plane geometry from Euclid's *Elements*: The *inscribed angle theorem* which is covered by two seperate theorems in the *Elements*.

At the top of the stamp we read in Portuguese *International Year of the Child 1979*.

**Inscribed angle theorem, part 1** Euclid *Elements* Book 3, Proposition 20

We draw a triangle in a circle with a chord as a base and the center of the circle as opposite vertex (point `O` on the stamp). We inscribe another triangle with the same base and the opposite vertex (point `A` on the stamp) on the arc which lies on the same side as `O` . **The angle of the triangle at **`O` is twice the angle at `A` ("`center` `angle` `=` `2·inscribed` `angle`").

**Inscribed angle theorem, part 2** Euclid *Elements* Book 3, Proposition 21

**All inscribed triangles as constructed in part 1 have equal inscribed angles in **`A` . The position of `A` has no effect on the angle at `A` .

Quite often, the two theorems are proved by first proving a special case (e.g. source). But this was not the way Euclid constructed the proof. More than 2300 years ago, he gave a simple and elegant proof in the *Elements*; it is illustrated in figure 1. This proof assumedly goes back to the Pythagoreans who worked on geometry 2500 years ago.

In figure 1 we find other labellings as in the stamp. The vertices in the center and on the arc are `M` and `C` . They are joined by a straight line. `AB` is the base of the triangles. `AMC` und `BMC` are isosceles triangles.

`δ`_{1} `+` `φ`_{1} = 180° and
`δ`_{1} `+` `2` `γ`_{1} = 180° ⇒ φ_{1} = 2 `γ`_{1}

In the same way we get `φ`_{2} `=` `2` `γ`_{2} , so `φ` `=` `2` `γ` . This derivation makes no use of the position of `C` , so both theorems are proved.

Figure 1 Copyright

Mathematics on the blackboard:

American Mathematical Society

Luca Pacioli (in German)

Euclid (in German)

Published 2019-08-27 last update 2019-08-26

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