Förstå randvinkelsatsen

Komplementvinkeln till centrumvinkeln ger att de två lika vinklarna i den likbenta triangeln är lika stora (180-(180-x)). Det ger att randvinkeln = halva centrumvinkeln vilket ger randvinkelsatsen. VSB.

Bild från GeoGebra

Inscribed angles where one chord is a diameter

texten från Wikipedia.

Let O be the center of a circle. Choose two points on the circle, and call them V and A. Draw line VO and extended past O so that it intersects the circle at point B which is diametrically opposite the point V. Draw an angle whose vertex is point V and whose sides pass through points A and B.

Angle BOA is a central angle; call it θ. Draw line OA. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ.

Angles BOA and AOV are supplementary. They add up to 180°, since line VB passing through O is a straight line. Therefore angle AOV measures 180° − θ.

It is known that the three angles of a triangle add up to 180°, and the three angles of triangle VOA are:

180° − θ

ψ

ψ.

Therefore

[math]\displaystyle{ 2 \psi + 180^\circ - \theta = 180^\circ. }[/math]

Subtract 180° from both sides,

[math]\displaystyle{ 2 \psi = \theta, \, }[/math]

where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB.