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Metric geometry : Angles on the circumference : Central and inscribed

angulo inscrito en una circunferenciaIn metric geometry measuring two concepts on which is based its axiomatic model: measures linear and angular measurements.
The linear measure abuts the Pythagorean theorem and relationship among these measures on Thales theorem.
The angular measurement from the relations expressed in a circle and with the above can describe the magnitude of geometric figures.
  • Central Angle -. It is that which has its apex at the center on the circumference and is understood as the arc.
  • Inscribed angle -. is one that has its vertex on the circle and its sides are chords.
angulos central e inscritos a circunferencia

Central and inscribed angles

An inscribed angle is half of the central angle covered by the same arc.

The sum of angles of a triangle is equal to two right, so the triangle CBP, that is isosceles (two equal angles) relationship will be fulfilled:

Angulos_internos_triangulo

So we have that

Angulo_llano

And consequently deduce that the central angle is twice the enrollee

Relacion_angulo_central_inscrito

It is easy to generalize this concept to positions of point P that are not so private, and we can decompose the angle in two and apply the same reasoning.

Relacion angulos central e inscrito

Relationship between Central and inscribed angles

For example, if the point P move along the circumference, the central angle is the sum of the two central angles that can decompose, thus being indifferent point position P.

Angulo_central_e_inscrito

Central angle and inscribed

Training

Ejemplos_angulos_inscritos

Examples inscribed angles

Metric geometry