inscribed angle

It is elementary and based on the theorem of an inscribed angle.

An inscribed angle is said to intersect an arc on the circle.

Hence, all inscribed angles that subtend the same arc (pink) are equal.

Note that the central angle for the golden inscribed angle is 360 2θ.

As another example, the inscribed angle theorem is the basis for several theorems related to the power of a point with respect to a circle.

An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red).

Therefore, the half of it (and thus the measure of the golden inscribed angle) is 180 θ.

An inscribed angle subtended by a diameter is a right angle (see Thales' theorem).

Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point, Thales' theorem.

This result may be extended to an arbitrarily inscribed angle by drawing a diameter from the apex of the angle.

Adding the two subangles again yields the result that the inscribed angle is half of the central angle.

This is a direct consequence of the inscribed angle theorem and the exterior angle theorem.

Typically, it is easiest to think of an inscribed angle as being defined by two chords of the circle sharing an endpoint.

The inscribed angle theorem is used in many proofs of elementary Euclidean geometry of the plane.

In that case, the hour lines are again spaced equally, but at double the usual angle, due to the geometrical inscribed angle theorem.

In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

The measure of the intercepted arc (equal to its central angle) is exactly twice the measure of the inscribed angle.

See inscribed angle, the proof of this theorem is quite similar to the proof of Thales' theorem given above.

The basic properties of inscribed angles are discussed in Book 3, Propositions 20-22 of Euclid's Elements.

In the simplest case, one leg of the inscribed angle is a diameter of the circle, i.e., passes through the center of the circle.

It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180 .

Thales' theorem is a special case of the inscribed angle theorem, and is mentioned and proved on the 33 proposition, third book of Euclid's Elements.

Draw lines VC and VD: angle DVC is an inscribed angle.

The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle.

Therefore triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ.