I'm Up and Down, and Round and Round Mind Map

A circle is the set of all points in a plane that are at the same distance from a fixed point called the centre. This fixed distance is called the radius.

When a radius is drawn to a point on the circle, immediately mark it equal to every other radius from the same centre.

A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation.

For symmetry questions, check whether the line passes through the centre. A chord not passing through the centre is not a line of symmetry of the whole circle.

A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.

In proof questions, write both facts: perpendicular from centre to chord and hence it bisects the chord. Do not skip the 90 degree condition.

One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.

In construction or proof, mention that the centre is equidistant from all three points because it lies on perpendicular bisectors.

The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.

Identify the same chord or same arc first. Then apply centre angle = 2 x circumference angle.

Angles in the same segment of a circle are equal. These angles stand on the same chord and have their vertices on the same arc segment.

Before using the theorem, circle the common chord and check that both angle vertices lie on the same side of that chord.

A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. In a cyclic quadrilateral, each pair of opposite angles is supplementary.

In a cyclic quadrilateral, mark opposite angle pairs with different symbols and write their sums separately.

Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.

To prove four points concyclic, try to form a quadrilateral and check one pair of opposite angles.