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This is a mind map about circles, the main contents include: arc length and sector area, regular polygons and circles, positional relationships between points and circles, straight lines and circles, and related properties of circles.

Edited at 2025-01-27 17:59:21

Mason·Carter

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Rumi 10 spiritual high-dimensional awakenings
Rumi: 10 dimensions of spiritual awakening. When you stop looking for yourself, you will find the entire universe because what you are looking for is also looking for you. Anything you do persevere every day can open a door to the depths of your spirit. In silence, I slipped into the secret realm, and I enjoyed everything to observe the magic around me, and didn't make any noise. Why do you like to crawl when you are born with wings? The soul has its own ears and can hear things that the mind cannot understand. Seek inward for the answer to everything, everything in the universe is in you. Lovers do not end up meeting somewhere, and there is no parting in this world. A wound is where light enters your heart.
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Mason·Carter

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Outline

Trigonometric functions
- 394
- 1
PlotWizard
proportional function
- 43
PlotWizard
Lines and angles
- 58
PlotWizard
quadratic radical
- 57
PlotWizard
Mathematics-Circle
- 36
PlotWizard
special parallelogram
- 21
Mason·Carter
round
- 23
EllaGrace
Equations of Lines and Circles
- 22
21_ethanSawyer
The concept of trigonometric functions
- 9
MM
The concept and properties of functions
- 20
Mason·Carter

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The properties of circles

definition

In a plane, the line segment OA rotates around one fixed endpoint O, and the figure formed by the other endpoint A is called a circle, and its fixed endpoint O is called the center of the circle. Line segment OA is called radius.

The line segment connecting any two points on a circle is called a string, and the string passing through the center of the circle is called a diameter.

The part between any two points in a circle is called an arc, abbreviated as an arc. The two endpoints of any diameter of the circle divide the circle into two arcs, and each arc is called a semicircle.

Two circles that can overlap are called equal circles.

Arcs that can overlap each other in the same circle or equal circle are called equal arcs.

nature

Pavement Theorem

Any straight line with a diameter is the axis of symmetry of a circle

Dividing bisects the string perpendicular to the diameter of the string and bisects the two arcs opposite the string

The diameter of the bisector (not the diameter) is perpendicular to the string and bisector the two arcs opposite the string

Arc, string, center angle

definition

The angle at which the vertex is at the center of the circle is called the center angle

nature

In the same circle or equal circle, the arcs opposite to the center angles of the equal circle are equal, and the strings opposite are equal.

In a same or equal circle, if the two arcs are equal, then the angles of the centers they are opposite are equal. The strings are equal.

In a same or equal circle, if the two strings are equal, then the center angles of the circle they are opposite are equal. The best and inferior arcs are equal respectively.

Circular angle

definition

The angle where the vertex is on a circle and both sides intersect the circle is called the circumferential angle

nature

The circumference angle to an arc is equal to half of the center angle to which it is

The circumference angles opposite to the same arc or equal arc are equal

It can be proved that the four points are in a circle

The circumference angle to which the semicircle (or diameter) is a right angle

The chord opposite the circumferential angle of 90° is the diameter

If all the vertices of a polygon are on the same circle, this polygon is called a circle inward quadrilateral, and this circle is called the polygon's outer circle.

The inline quadrilaterals of the circle complement each other

It can be proved that the four points are in a circle

Points and circles, lines and circles

Points and circles

The outside of the circle is equivalent to d>r

In a circle, it is equivalent to d=r

In circle is equivalent to d<r

Three different points determine a circle

Counter-proof law

Lines and circles

Intersecting with circles is equivalent to d<r

The tangent to the circle is equivalent to d=r

determination

The straight line passing through the outer end of the radius and perpendicular to this radius is a tangent of a circle

nature

The tangent of a circle is perpendicular to the radius of the over-tangent point

From a point outside the circle, you can lead two tangents of the circle. Their tangents are equal in length, which divides the angle between the two tangents with the connecting line of the center of the circle.

The circle that is tangent to each side of a triangle is called the intangible circle of a triangle; the center of the intangible circle is the intersection of the bisector of the triangle, called the inner center of the triangle

r=2S/(a b c)

The separation from the circle is equivalent to d>r

Regular polygons and circles

The center of the circumferential circle of a polygon is called the center of the regular polygon; the radius of the circumferential circle is called the radius of the regular polygon; the center angle opposite to each side of the regular polygon is called the center angle of the regular polygon; the distance from the center to one side of the regular polygon is called the regular polygon The edge distance

Arc length and sector area

definition

The figure consisting of the arcs opposite the two radii and the center angle of the circle are called sectors

The line segment connecting the cone vertex and any point on the circumference of the bottom surface is called the cone bus line.

formula

Arc length

l=nΠr/180

sector

S=nΠr^2/360=rl/2

Cone surface area

S=Πrl Πr^2=Πr(r l)