What Is a Tangent Line?

Geometrically, a tangent refers to a straight line that just touches a point on a curve. More precisely, when a tangent passes a point (ie, a tangent point) on a curve, the direction of the tangent is the same as that of the point on the curve. In plane geometry, a line that has only one common intersection with a circle is called a tangent to the circle. ^[1]

P and Q are two adjacent points on the curve C, and P is a fixed point. When the Q point approaches the P point infinitely along the curve C, the secant PQ

in

Tangent property theorem

The tangent of the circle is perpendicular to the radius passing through the tangent point. ^[2]

Corollary 1 : A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.

Corollary 2 : A straight line passing through the tangent point and perpendicular to the tangent line must pass through the center of the circle.

Tangent main properties

The line segment DA is perpendicular to the line AB (AD is the diameter)

(1) Tangent and circle have only one common point;

(2) The distance between the tangent and the center of the circle is equal to the radius of the circle;

(3) The tangent line is perpendicular to the radius passing through the tangent point;

(4) A straight line passing through the center of the circle perpendicular to the tangent line must pass through the tangent point;

(5) The line passing through the tangent point perpendicular to the tangent line must pass through the center of the circle;

(6) The tangent and secant that draw a circle from a point outside the circle. The length of the tangent is the middle of the ratio of the length of the two line segments from this point to the intersection of the secant and the circle.

Among them, (1) is obtained from the definition of tangent line, (2) is obtained from the theorem of the position relationship between straight lines and circles, and (6) is derived from similar triangles, which is the cut line theorem.

Tangent determination and properties

Judgment Theorem of Tangent Line: A straight line passing through the outer end of a radius and perpendicular to this radius is a tangent to a circle. The tangent of the circle is perpendicular to the radius of the circle's tangent point.

Geometric language: lOA, point A is on O

Line l is a tangent to O (tangent judgment theorem)

Theorem of the property of tangent line: The tangent line of the circle is perpendicular to the radius passing through the tangent point.

Geometric language: OA is the radius of O, the line l cuts O at point A

l OA (tangent property theorem)

Corollary 1 The diameter passing through the center of the circle and perpendicular to the tangent must pass through the tangent point,

Corollary 2 A line passing through the tangent point and perpendicular to the tangent line must pass through the center of the circle.

Tangent Tangent Length Theorem

Theorem: Two tangent lines of the circle can be drawn from a point outside the circle. Their tangent lines are equal in length. ^[3]

Geometric language: chord PB, PD cut O at A, C

PA = PC, APO = CPO (Tangent Length Theorem)

Chord angle

The chord angle theorem : The chord angle is equal to the circumferential angle of the arc pair it sandwiches.

Geometric language: BCN is what A is for

BCN = A

Corollary: If the arcs between the two chord cut angles are equal, then the two chord cut angles are also equal.

Chord-cut angle concept: The vertex is on a circle, and the angle where one side intersects the circle and the other side is tangent to the circle is called a chord-angle. It is the third kind of circle-related angle after the circle center angle and the circle angle. This angle must satisfy three conditions:

(1) The vertex is on the circle, that is, the vertex of the corner is the tangent of a tangent to the circle;

(2) One side of the corner intersects the circle, that is, one side of the corner is the ray of a chord crossing the tangent point;

(3) The other side of the corner is tangent to the circle, that is, the other side of the corner is a ray on the tangent line with the tangent point as the endpoint. They are the criteria for judging whether an angle is a chord-cut angle. In the figure, they are not chord cut angles;

(4) The chord cut angle can be regarded as a special case of the circumferential angle, that is, the angle formed when one side of the circumferential angle is rotated around the vertex to be tangent to the circle. Because of this, the chord cut angle has similar properties to the circumferential angle.

The chord tangent angle theorem: The chord tangent angle is equal to the circumferential angle of the arc pair that it sandwiches. It is one of the important theorems in a circle to prove that the angles are equal. ^[4]

The cutting line theorem: a tangent and a secant that draw a circle from a point outside the circle. The length of the tangent is the middle term of the ratio of the length of the two line segments from this point to the intersection of the secant and the circle.

Corollary: The product of the length of the two secant lines leading from a point outside the circle to the point where each secant line intersects with the circle is equal.