Circle

• A circle $◯$ is the locus of points in a plane that are at a fixed distance $r$ (the radius) from a fixed point $O$ (the center of the circle).

  • The circumference $C=2\pi r=\pi d$ is the length of one circuit around the circle
  • The area enclosed by a circle is equal $A=\pi r^2$.
  • A diameter is a line segment whose endpoints are on the circle and passes through the center of the circle.
    • The diameter $d=2r$ is the length of any diameter.
  • A chord is a line segment whose endpoints are both on the circle.
    • Every chord in the circle is inside the circle.
    • The longest chords in a circle are the diameters.
    • Let $AB$ and $A^{\prime }{B}^{\prime }$ be two chords in the circle. Then $\overline{AB}=\overline{A^{\prime }{B}^{\prime }}$ if and only if $\angle AOB=\angle A^{\prime }O{B}^{\prime }$
  • A secant is a line that intersects the circle at two points.
  • A tangent is a line that intersects the circle at exactly one point.
    • A tangent to the circle is perpendicular to the radius to the tangent point
    • A tangent to the circle is a straight line passing through a point on a circle, and perpendicular to the radius to this point, is a tangent to the circle
  • A radius (plural: radii) is a line segment whose endpoints are the center of the circle and a point on the circle.
  • A (circular) arc is the arc of a circle between two distinct points on the circle.
    • Arc Length $s=r\theta$
  • A segment of a circle is a region bounded by a chord and its minor (or major) arc.
  • A sector of a circle is a region bounded by two radii of the circle and an arc intercepted by those radii.
    • Area of a sector of circle $\frac{\theta }{360^{\circ }}\cdot \pi r^2$

• An inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle

  • (Inscribed angle theorem) An angle $\theta$ inscribed in a circle is half of the central angle $2\theta$ that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.
    • (Thales’s theorem) if $A$, $B$, and $C$ are distinct points on a circle where the line $AC$ is a diameter, the angle $\angle ABC$ is a right angle.

• A central angle is an angle whose vertex is the center of the circle and whose sides are radii of the circle.

  • The sum of two inscribed angle subtends on the same chord on both sides is $180^{\circ }$

• A straight line and a circle intersect in at most two points.

• Every triangle have a unique circumcircle

• A square can be circumscribed in a circle if and only if the sum of each pair of opposite angles in it is $180^{\circ }$.

• Inside each triangle has a unique circumscribed circle

• The center of the circumscribed circle is the point of intersection of the triangle’s perpendicular bisectors.

• The locus of points in triangle $△ABC$ that are equidistant from sides $AB$ and $AC$ is the bisector of the angle $\angle BAC$.

• Let $△ABC$ be a triangle whose area is $S$, perimeter is $k$, and the radius of the inscribed circle is $r$. The following exists: $S=\frac{k\cdot r}{2}$