Förstå randvinkelsatsen - Versionshistorik

Hakan: /* Inscribed angles where one chord is a diameter */

2012-03-21T11:19:10Z

Inscribed angles where one chord is a diameter

← Äldre version		Versionen från 21 mars 2012 kl. 11.19
Rad 22:		Rad 22:
	Therefore		Therefore

	:ψ + 180<sup>o</sup> - θ = 180<sup>o</sup>		:2ψ + 180<sup>o</sup> - θ = 180<sup>o</sup>

	Subtract 180° from both sides,		Subtract 180° from both sides,

Hakan: /* Inscribed angles where one chord is a diameter */

2012-03-21T11:17:49Z

Inscribed angles where one chord is a diameter

← Äldre version		Versionen från 21 mars 2012 kl. 11.17
Rad 12:		Rad 12:
	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''.		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 angle~~\|supplementary~~. They add up to 180°, since line ''VB'' passing through ''O'' is a straight line. Therefore angle ''AOV'' measures 180° − θ.		Angles ''BOA'' and ''AOV'' are supplementary angle. 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:		It is known that the three angles of a triangle add up to 180°, and the three angles of triangle ''VOA'' are:

Hakan: /* Inscribed angles where one chord is a diameter */

2012-03-21T10:54:01Z

Inscribed angles where one chord is a diameter

← Äldre version		Versionen från 21 mars 2012 kl. 10.54
Rad 8:		Rad 8:
	''texten från [http://en.wikipedia.org/w/index.php?title=Inscribed_angle&action=edit&section=3 Wikipedia.]''		''texten från [http://en.wikipedia.org/w/index.php?title=Inscribed_angle&action=edit&section=3 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''.		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 ~~[[radius\|~~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''.		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 angle\|supplementary. They add up to 180°, since line ''VB'' passing through ''O'' is a straight line. Therefore angle ''AOV'' measures 180° − θ.		Angles ''BOA'' and ''AOV'' are supplementary angle\|supplementary. They add up to 180°, since line ''VB'' passing through ''O'' is a straight line. Therefore angle ''AOV'' measures 180° − θ.

Hakan: /* Inscribed angles where one chord is a diameter */

2012-03-21T10:51:59Z

Inscribed angles where one chord is a diameter

← Äldre version		Versionen från 21 mars 2012 kl. 10.51
Rad 10:		Rad 10:
	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''.		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 [[radius\|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''.		Angle ''BOA'' is a central angle; call it ''θ''. Draw line ''OA''. Lines ''OV'' and ''OA'' are both [[radius\|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 angle\|supplementary]]. They add up to 180°, since line ''VB'' passing through ''O'' is a straight line. Therefore angle ''AOV'' measures 180° − θ.		Angles ''BOA'' and ''AOV'' are supplementary angle\|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:		It is known that the three angles of a triangle add up to 180°, and the three angles of triangle ''VOA'' are:

	: 180° − θ		: 180° − θ
Rad 22:		Rad 22:
	Therefore		Therefore

	:<~~math~~> ~~2 \psi + 180^\circ~~ - ~~\theta~~ = 180~~^\circ.~~ </~~math~~>		:ψ + 180<sup>o</sup> - θ = 180<sup>o</sup>

	Subtract 180° from both sides,		Subtract 180° from both sides,

	:~~<math>~~ 2 ~~\psi~~ = ~~\theta, \,</math>~~		:2 ψ = θ

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

Hakan: /* Inscribed angles where one chord is a diameter */

2012-03-20T22:18:25Z

Inscribed angles where one chord is a diameter

← Äldre version		Versionen från 20 mars 2012 kl. 22.18
Rad 5:		Rad 5:
	===Inscribed angles where one chord is a diameter===		===Inscribed angles where one chord is a diameter===
	[[Image:InscribedAngle 1ChordDiam.svg\|right]]		[[Image:InscribedAngle 1ChordDiam.svg\|right]]

			''texten från [http://en.wikipedia.org/w/index.php?title=Inscribed_angle&action=edit&section=3 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''.		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''.

Hakan den 20 mars 2012 kl. 22.17

2012-03-20T22:17:08Z

← Äldre version		Versionen från 20 mars 2012 kl. 22.17
Rad 2:		Rad 2:

	Bild från GeoGebra		Bild från GeoGebra

			===Inscribed angles where one chord is a diameter===
			[[Image:InscribedAngle 1ChordDiam.svg\|right]]
			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 [[radius\|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 angle\|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> 2 \psi + 180^\circ - \theta = 180^\circ. </math>

			Subtract 180° from both sides,

			:<math> 2 \psi = \theta, \,</math>

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

Hakan: Skapade sidan med '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 vilk...'

2012-03-20T22:03:56Z

Skapade sidan med '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 vilk...'

Ny sida

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