The counter proof method is used to prove the theorem of the nature of tangents

Remember that the hypothesis must be wrong, you should do it correctly first, and if the hypothesis does not hold after the contradiction, then the original question will naturally be established, which is the mystery of the counter-proof method).

I'll use the counter-evidence method to prove it, and I'll use what you gave.

First, a straight line that passes through the outer end of the radius and is perpendicular to this radius is a tangent of the circle.

The reference book must be written like this, assuming first that AT is not a tangent of a circle.

Since it is assumed that at is not a tangent of a circle, then oam ≠ 90° and the new om at, get oma=90°, so in rt aot (the assumed right angle is oma, although it cannot be seen from the figure), the hypotenuse is the largest, so oa of oma pairs is > om, and oa is the radius of the circle, m is a point outside the circle, and the distance from the point outside the circle to the center of the circle must be less than the distance from the point on the circle to the center of the circle (common sense), so oa>om is not established, and the contradiction is introduced, Therefore, the original hypothesis is not valid, and the original question is the true proposition.

Now this graph at is a tangent oa is the radius of the tangent point by the tangent property theorem at is perpendicular oa but now oa and at don't admit that they are perpendicular we have to prove that they are perpendicular okay since at and oa you don't recognize perpendicular that is, from o point to at at this straight line as a perpendicular line a is not a perpendicular foot then i will find another m point to do the perpendicular foot there is om perpendicular at (because a point must be a vertical foot, so m point must not be a vertical foot we just find a contradiction Force point A to admit that he is a vertical foot) Since OM is perpendicular to AT, then the distance from OM to AT must be the shortest OA does not admit that it is a perpendicular line, its length must be greater than OM and OA is a radius OM is less than the radius AT should intersect with the circle and AT is tangent to the circle This is a contradiction, so OA and AT cannot be fooled They are vertical relations and conclude that the tangent of the circle is perpendicular to the radius of the tangent point.

The proof of the theorem of the nature of tangents is as follows:

1. Determination and nature of tangents.

1. The determination theorem of tangents: a straight line that passes through the outer end of the radius and is perpendicular to this radius is a tangent of a circle. The tangent of the circle is perpendicular to the radius of this circle.

Geometric language: l oa, point a on o.

The straight line l is the tangent of o (tangent decision theorem).

2. The nature of the tangent theoremThe tangent of the circle is perpendicular to the radius of the tangent point.

Geometric language: oa is the radius of o, and the straight line l tangentiates o to the point a

l OA (Tangent Property Theorem).

Corollary 1: The diameter that passes through the center of the circle and perpendicular to the tangent must pass through the tangent point.

Corollary 2: A straight line that passes through the tangent and is perpendicular to the tangent must pass through the center of the circle.

3. Tangent length theorem.

Theorem: Two tangents of a circle drawn from a point outside the circle are of equal length, and the center of the circle and the line connecting this point divide the angle between the two tangents.

Geometric language: The strings pb and pd are cut to a and c.

Pa = PC, APO = CPO (tangent length theorem).

4. Chord cut angle.

The chord chamfer theorem: The chord chamfer is equal to the circumferential ridge angle of the pair of arcs it is sandwiched.

Geometric language: What BCN is sandwiched, A is right.

bcn=∠a

It is inferred that if the two chord tangent angles are sandwiched by the arc phase of the cherry blossoms, etc., then the two chord tangent angles are also equal.

Geometric language: What BCN is sandwiched, ACM is right.

bcn=∠acm

The concept of chord tangent angle: the vertex is on the circle, one side intersects the circle, and the other side is tangent to the circle is called the chord tangent angle, which is the third kind of angle related to the circle after the central angle and the circumferential angle, and this angle must meet three conditions:

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

2) One side of the angle intersects the circle, i.e., one side of the angle is the ray where one of the strings of the tangent point is located;

3) The other side of the angle is tangent to the circle, that is, the other side of the angle is a ray on the tangent with the tangent as the endpoint.

Proof: Let Sun Pai only have a straight line B, A is not in the plane, and B is in the plane of Heshan.

Suppose that if a line outside the plane is parallel to a line in the plane, then the line is not necessarily parallel to the plane.

If the line a is not parallel to the plane, and since a is not in the plane, then a intersects with , let a = f

The crossing point f makes a straight line c b in the plane, and since a b then a c

and f a, and f c, i.e., a c = f, which contradicts a c. So the assumption is incorrect, the original proposition is correct.

It is known that oa is the radius of the circle o, and ab is a cut line of the circle o.

Verify: OA AB. Proof:

If oa ab is not true, then o can be used as oc ab to c, because a is the only point on ab that intersects the circle, c must be outside the circle, and in the frank line connecting a point outside the line and the straight line, the perpendicular segment is the shortest, so oc< OA, which contradicts C outside the circle, so it is impossible to assume that there must be OA AB.

The matching andan determination theorem of tangents.

Derive theorem: According to "the tangent of the line and o d=r".

Because d=r is tangent to o, where d is the distance from the center of the circle o to the line, i.e., perpendicular, and by d=r, we can get the outer end of the radius r, that is, the end of the radius oa, and the determination theorem of the tangent can be obtained:

A straight line that passes through the outer end of a radius and is perpendicular to this radius is a tangent of a circle Analysis: 1 How many straight lines are there perpendicular to a radius?

2. How many perpendiculars of the radius can be made through the outer end of the radius?

3. What happens if you remove the "pass through the outer end of the radius" in the theorem? What about removing "perpendicular to radius"?

Thought 1: According to the decision theorem above, what conditions need to be met to prove that a straight line is a tangent of o?

Summary: This line has a common point with o; The radius of the point is perpendicular to this line Thought 2: How many ways can a straight line be proved that a straight line is a tangent of a circle?

A straight line with only one common point is a tangent of a circle.

A straight line whose distance to the center of the circle is equal to the radius is the tangent of the circle.

The decision theorem above.

The nature of the tangent is:

1. Tangents and circles 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 is perpendicular to the radius of the tangent point.

4. A straight line perpendicular to the tangent line through the center of the circle must pass the tangent point.

5. The straight line perpendicular to the tangent line through the tangent point must pass through the filial piety hail tomb in the center of the circle.

6. The tangent and secant line of the circle are drawn from the outer point of the circle, and the tangent length is the middle term of the ratio of the length of the two line segments from this point to the intersection of the secant line and the circle.

The theorem of the tangent is that a straight line passing through the outer end of the radius and perpendicular to this radius is a tangent of a circle, and the tangent of a circle is perpendicular to the radius of the circle past the tangent point.

Tangent. <>

A tangent is a straight line that touches a point on a curve. More precisely, when a tangent passes through a point on a curve (i.e., a tangent), the tangent is in the same direction as the point on the curve, and the part of the tangent near the tangent is closest to the part of the curve near the tangent.

Geometric definition: p and q are two adjacent points on the curve c, p is the fixed point, when the q point is infinitely close to the p point along the curve c, the limit position pt of the secant pq is called the tangent of the curve c at the point p, and the p point is called the tangent point; The straight line pn that passes through the tangent point p and is perpendicular to the tangent pt is called the normal of the curve c at the point p.

Note: In plane geometry, a straight line with a common intersection point with a circle is called a tangent of a circle, which is not applicable to general curves. pt is the tangent of the curve c at the point p, but it has another intersection with the curve c; Conversely, the straight line l, although it has only one intersection with curve C, is not a tangent of curve C.

The "counter-proof method" is used to prove that there are three steps: (1) assuming that the tangent at is not perpendicular to the radius oa of the tangent point, (2) making a perpendicular line of at the same time om is contradictory by proof, om oa The radius has a quantitative relation in the positional relationship between the line and the circle, and the intersection of at and o contradicts the problem (3) Acknowledging the conclusion at ao The property theorem of the tangent is that the tangent of the circle is perpendicular to the radius of the tangent point

Dear, if you are satisfied, please help me like it, thank you.

Proof with the "Counter-Evidence Method" is a three-step process:

1) Assuming that the tangent at is not perpendicular to the radius oa of the tangent point, (2) making a perpendicular line of at at the same time om is contradicted by proof, and the radius of om oa has a quantitative relation in the positional relationship between the line and the circle, and the intersection of at and o contradicts the problem

3) Acknowledging the desired conclusion at ao

Theorem of the nature of tangents: The tangent of a circle is perpendicular to the radius passing through the tangent point