Circle

• A circle is the locus of points in a plane that are at a fixed distance from a given point.

• Circumference

• Central angle

• Arc

• Chord

• Diameters

• Inscribed Angle

• incircle or inscribed circle (מעגל חסום)

• Circumscribed circle (or circumcircle) (מעגל חוסם)

• A locus is the set of all points and only those points that satisfy a given condition (or set of conditions)

• A segment of a circle is a region bounded by a chord and its minor (or major) arc.

• Area of a circle $\pi r^2$

• The circumference $2\pi r=\pi d$

• Arc

  • Arc Length $s=r\theta$

• Chord

  • 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 $\angle AOB=\angle A^{\prime }O{B}^{\prime }$

• Inscribed Angle

  • 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.
  • 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.

• 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

• Every triangle have a unique circumcircle

• circumcenter of triangle is the point where three perpendicular bisectors from the sides of a triangle intersect or meet

• 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: $S=\frac{k\cdot r}{2}$

• 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 $\frac{\theta }{360^{\circ }}\cdot \pi r^2$