Barrel arc

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Sketch for the pair of barrel arcs

The barrel arc is a term from geometry . It denotes an arc of a circle over a line from which this line always appears at the same angle. Since there is such a circular arc on each side of the given route, one speaks of a barrel arc pair .

Definition and application

The barrel arc can also be viewed as the geometric location ( set ) of all points from which a chord of the circle can always be seen at the same angle. This particular angle is also called the circumference angle .

A special case of the barrel arc is the Thales semicircle . In this, each circumferential angle measures 90 °, and the chord is the diameter of the circle.

The barrel arc helps prove a number of geometric theorems. For example, the following statement can be proven well with the barrel arc:

In a triangle, the normal perpendicular to one side and the bisector of the opposite angle of the side intersect on the circumference ( south pole set ).

The terms barrel arc and barrel circle are used synonymously.

See also: circle angle

construction

Barrel arc - start of construction with tendon. The individual steps are numbered.

example 1

  1. Draw string (AB),
  2. Create the desired barrel angle ( ) at a tendon end point (for example here at A) .
  3. In the vertex (at A), make the perpendicular on the angle leg.
  4. Construct the perpendicular for (AB).

The intersection of the vertical and perpendicular results in the center point of the barrel circle (M) (see picture).

Example 2

  1. Draw string (AB),
  2. Subtract the circumferential angle ( ) from 90 ° (for an obtuse angle: subtract 90 ° from the circumferential angle) and apply this angle to A and B.
  3. The intersection of the two free legs is the center of the circumference.

Example 3

Barrel arc - start of construction with the circumferential angle
  1. Draw a (circumferential) angle ( ) with point C
  2. Pick a point A on one leg and draw a circle with radius | AB | around A, this cuts the other leg in B.
  3. Erect the vertical verticals on AB and AC, these intersect in the center of the barrel circle.

literature

  • Günther Aumann: Circular Geometry: An Elementary Introduction . Springer, 2015, ISBN 9783662453063 , pp. 16-17
  • Rolf Baumann: More success in mathematics: 8th grade geometry . Mentor, 2008, ISBN 9783580656294 , pp. 78-80
  • Siegfried Krauter, Christine Bescherer: The Elementary Geometry Experience: A workbook for independent and active discovery . Springer, 2012, ISBN 9783827430250 , pp. 74-76

Web links