円 — 部分と定理 ⟦number⟧

The wheel that changed the world

• The circle is the most perfect shape in nature — every point on its edge is the same distance from the centre.
• Ancient mathematicians discovered remarkable properties hidden in this simple shape.

The parts of a circle

• Radius 半径: centre to edge. Diameter 直径: right across ($= 2r$). Circumference 圆周: the distance round.
• Chord 弦: a line joining two points on the circle. Tangent 切线: touches at exactly one point.
• Arc 弧: part of the circumference. Sector 扇形: a slice between two radii. Segment 弓形: the region cut off by a chord.

Circle theorems (Core)

• The angle in a semicircle is $90^{\circ}$ (an angle standing on a diameter).
• The angle between a tangent and a radius is $90^{\circ}$.

The main parts of a circle; a chord joins two points, a tangent touches at just one

Circle theorems (Extended)

• The angle at the centre is twice the angle at the circumference (same arc).
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral 圆内接四边形 add up to $180^{\circ}$.

The angle at the centre is always twice the angle at the circumference for the same arc.

Worked example

• Angle at circumference $= 40^{\circ}$ → angle at centre $= 2 \times 40 = 80^{\circ}$.
• In a cyclic quadrilateral, one angle $= 85^{\circ}$ → opposite angle $= 180 - 85 = 95^{\circ}$.

A sector is a slice between two radii; a segment is cut off by a chord; an arc is part of the circumference

The missing Extended circle facts

• Alternate segment theorem: the angle between a tangent and a chord equals the angle at the circumference in the opposite segment. If that tangent-chord angle is $52^{\circ}$, the matching circumference angle is $52^{\circ}$.
• The perpendicular bisector of a chord passes through the centre; equal chords are equally far from it. Tangents PA and PB from the same outside point P have equal lengths, so $PA=7$ cm implies $PB=7$ cm.

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