G.9 Geometry

🔹 1. Theorem: Angle at the Center

Statement:
The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the remaining part of the circle.

• Symbolically:
  If arc $AB$ subtends $\angle AOB$ at the center and $\angle ACB$ at any point on the circle, then:

  $$\angle AOB=2\angle ACB$$

🔹 2. Theorem: Angles in the Same Segment

Statement:
Angles subtended by the same arc in the same segment are equal.

• Example:
  If points $A,B,C,D$ lie on a circle and $\angle ACB$ and $\angle ADB$ are on the same arc $AB$, then:

  $$\angle ACB=\angle ADB$$

🔹 3. Theorem: Angle in a Semicircle

Statement:
The angle in a semicircle is a right angle.

• Meaning:
  If $AB$ is the diameter and $C$ is any point on the semicircle, then:

  ∠ACB=90∘\angle ACB = 90^\circ∠ACB=90∘

🔹 4. Theorem: Cyclic Quadrilateral

Statement:
The opposite angles of a cyclic quadrilateral (all vertices lie on the circle) are supplementary.

• That is:

  ∠A+∠C=180∘,∠B+∠D=180∘\angle A + \angle C = 180^\circ,\quad \angle B + \angle D = 180^\circ∠A+∠C=180∘,∠B+∠D=180∘

🔹 5. Theorem: Tangent and Radius

Statement:
A tangent to a circle is perpendicular to the radius at the point of contact.

• That is:
  If $T$ is the point of tangency, then:

  $$OT\perp \text{tangent line at }T$$

🔹 6. Theorem: Lengths of Tangents from a Point

Statement:
Tangents drawn from an external point to a circle are equal in length.

• That is:
  If $PA$ and $PB$ are tangents from point $P$, then:

  $$PA=PB$$