Manfred Börgens
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Kreiswinkelsatz auf Tafel

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.

δ1 + φ1 = 180°   and   δ1 + 2 γ1 = 180°   ⇒   φ1 = 2 γ1

In the same way we get  φ2 = 2 γ2 , 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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