Please prove the circular power theorem by analytical method;
Theorem 1. 1 Any point leads a straight line and a circle to intersect at two points (tangent coincidence), then it is constant, and
( 1. 1)
Where is the radius of the circle and the distance from the center of the circle.
The voucher is shown in figure 1- 1:
Without losing generality, let the center of the circle be at the origin of the plane rectangular coordinate system with radius, and its equation is:
( 1.2)
And the coordinate of this point is set to, then the equation is:
( 1.3)
or
( 1.4)
(Formula (1.3) represents a straight line, and formula (1.4) represents a straight line.)
Let the coordinates of the intersection of a straight line and a circle be:
There are two situations as follows:
Case 1: simultaneous (1.2) and (1.3) elimination, quadratic equation standardization;
( 1.5)
For two equations (1.5), according to Vieta's theorem:
According to the distance formula and relationship between two points, there are:
(as a fixed value).
Case 2: Simultaneous (1.2), (1.4), the coordinates of A and B are:
Can also be calculated (as a fixed value)
To sum up: (as a fixed value), in which
Notes are shown in figure 1-2. If it is outside the circle, the power of the circle is equal to the square of the line segment tangent to the circle. This is the conclusion of the tangent theorem, namely:
( 1.5)
As shown in figure 1-3, if it is in a circle, the power of the circle is equal to the square of half of the intersecting and vertical chords. This is the conclusion of the symphony theorem, namely:
( 1.6)
Question 2 (idempotent line) has a circle. If the powers of a pair of points are equal, a point is said to be an idempotent point of two circles. The set consisting of all idempotent points of two circles can be a straight line, a circle or an empty set. Please discuss it according to the situation.
Say that the idempotent point set of two circles is the idempotent set or idempotent line of two circles. On idempotent sets, the following five propositions are given:
Proposition 2. 1 If the distances between the radii and the centers of two circles are respectively, then its idempotent set is a straight line.
This proof does not lose generality. Let the center of the circle be, the radius be, and the coordinates of the point be. Because of the propositional conditions. As shown in Figure 2. 1:
According to the definition of cyclic power, there are:
The power of the circle:
The power of the circle:
These two forces are equal, so:
Absolute value removal has two results:
or
Expand and organize:
(2. 1)
or
(2.2)
Because, so:
Equation (2.2) is contradictory and is omitted.
It is verified that equation (2. 1) is an idempotent equation of two circles and a straight line.
Proposition 2.2 If the radius and center distance of two circles are zero, then its idempotent set is a straight line and a circle, and the center of this circle is at the midpoint of the connecting line.
It is proved that equation (2.2) has a solution and the solution is a circle. It is verified that the lines and circles represented by equations (2. 1) and (2.2) are included in the idempotent set.
According to Equation (2.2), the coordinate of the center of the circle is, which is exactly the midpoint of the two centers.
In particular, the commentary points out that when the circle represented by Equation (2.2) degenerates to a point, this point is at the midpoint of the connecting line.
Proposition 2.3 If two circles with different radii are concentric, their idempotent sets are circles.
It is proved that equation (2. 1) has no solution, because. However, it is verified that the circle represented by (2.2) is contained in the idempotent set.
The straight line represented by equation (2. 1) is called the root axis or idempotent axis. The circle represented by Equation (2.2) is called an idempotent circle.
Proposition 2.4 The root axis of two circles is perpendicular to the connecting line of two circles.
It is proved that the straight line represented by equation (2. 1) is vertical, so it is perpendicular to the connecting line.
Proposition 2.5 Two circles intersect (or tangent), and the root axis passes through two intersection points (or tangents).
Prove that as shown in Figure 2-2, the equation for setting a circle is as follows:
(2.3)
(2.4)
Among them, according to the intersection or tangency conditions, there are:
(2.5)
(2.4)-(2.3):
Get (2.6)
Substituting the above formula into (2.3) and (2.4) verifies that there is a real root, so it is the abscissa of the two intersections (A and B coincide when they are tangent). The comparison formula (2. 1) shows that the root axis passes through this point, and the proposition is proved.
The geometric meaning of comment inequality (2.5) is:
The left equal sign holds, indicating that two circles are inscribed;
The edge equal sign holds, which means that the two circles are circumscribed;
The rest represent the intersection of two circles.
Question 3 (Arithmetic Power Line Theorem) If there are four points in Figure 3- 1, the necessary and sufficient condition is (3. 1).
It is proved that as shown in Figure 3-2, circles are made with the center and radius respectively, so that they are idempotent to A .. This shows that:
(3.2)
If, because it is the idempotent point of two circles, the straight line is the root axis of two circles, so it is the idempotent point of two circles. So:
(3.3)
Simultaneous (3.2)(3.2) and elimination (3. 1) proved the necessity.
If (3. 1) holds, simultaneous (3. 1) and (3.2) can get (3.3), so it is the idempotent point of two circles, so the sufficiency holds.
The conclusion of this topic is called "arithmetic power line theorem". In the process of proof, there is no problem that there is no absolute value on both sides of (3.2) and (3.3). This is because every point on the root axis is either outside, inside or at the intersection of two circles. In any case, excluding the absolute value of the circular power formula is in the form of (3.2) and (3.3). In other words, the idempotent equation of any point on the root axis of two circles can be written as:
We can also notice this by carefully understanding the proof of proposition 2. 1.
Inference 3. 1 It is known that the locus of a point whose square difference between two points is constant is a straight line perpendicular to these two points.
Question 4: As shown in Figure 4- 1, the centers of three circles are not a line. Two common tangents of a circle, two common tangents of a circle, two common tangents of a circle. Where they are their midpoints. Verification:
Three straight lines intersect at one point.
Evidence is known by conditions:
A straight line is the root axis of a circle,
A straight line is the root axis of a circle,
A straight line is the root axis of a circle.
Let and intersect, then it is the idempotent point of three circles, that is to say, it is on the root axis of the circle, so these three lines are * * * points.
The common tangent of this topic can be circumscribed or inscribed. At the same time, the tangent condition limits that three circles cannot contain each other and can only be separated or intersected (including tangency). In fact, the following proposition explains the positional relationship of the root axes of three circles more essentially:
Proposition 4. 1 Three circles are not concentric, the center of the circle is not a line, and the two axes intersect at one point; Three circles are not concentric, the center is a line, and the two axes are parallel.
It is deduced that three circles with different concentric lines have one and only one idempotent point, which is called the root center of three circles.
Question 5: As shown in Figure 5- 1, the tangent of the circle, the tangent point is, the passing point is a straight line intersecting the circle, and the chord intersects the point. Verification: (5. 1)
Proof as shown in figure 5-2, connection.
According to the circular power theorem:
(5.2)
(5.3)
Because the circle is on the tangent, therefore, according to the arithmetic power line theorem, there are:
Merge (5.2) and (5.3):
Comment on the conclusion of this question and the power theorem of plane geometric circle (2)
Question 4 in is equivalent and is deduced as follows:
As shown in Figure 6- 1, the circumscribed circle of a circle is tangent to a point, intersects an edge at a point, and intersects an edge. The tangent through the point, the tangent point is, connect, intersect. Proof: equal to the tangent length of the circle passing through this point.
Prove that as shown in Figure 6-2, let point A be the tangent of two circles and connect the two circles.
According to the tangent angle theorem:
So:
So: * * circle.
So:
According to the tangent angle theorem:
So:
The complementary angles of equal angles are equal, so:
There are:
therefore
Is the power of the circle, so it is equal to the tangent length of the circle passing through this point.
Comment 6. 1 Pay attention to Figure 6-3 and Figure 6-4. Two circles are tangent to a point, and the straight lines passing through the point intersect respectively, and then the straight lines intersect respectively.
The above conclusions are the same for internal cutting and external cutting, please prove them separately.
In question 6, connection can prove that this is another way to solve question 6. (Please try)
Note 6.2 is shown in Figure 6-4, which is different from Figure 6- 1. Tangent line and circle are tangent to another point l', which are different from each other. It is connected and expanded, and intersects with the extension line of. Is this conclusion still valid? Is it equal to the tangent length of the circle?
The answer is yes, please prove it.
Question 7 (20 16 National Senior High School Mathematics League) is shown in Figure 7- 1, which is located on the extension line of. Set the epicenter at and the straight lines intersect at. Proof: It is an isosceles triangle.
Prove that as shown in Figure 7-2, let circles intersect at points and connecting lines intersect at.
The straight line is the root axis of the circle, so the point E to the circle is equal to the power of the circle.
So:
(7. 1)
and
(7.2)
So split it equally.
At the same time, from the nature of the root axis
It can be judged from this
So it is an isosceles triangle.
Explain that this topic uses the nature of the bisector of the triangle inner angle, which can be described as:
In proposition 7. 1, on the line segment, then it is equally divided if and only if.
The proof is shown in Figure 7-2, where the area is represented by the area formula:
Divide by the above two formulas.
(7.3)
because