It is proved that the line connecting the two tangent points must pass through a certain point Such as the title

It is proved that the line connecting the two tangent points must pass through a certain point Such as the title

Take the coordinate system as shown in the figure, circle X & amp; sup2; + Y & amp; sup2; = R & amp; sup2;. Straight line x = a
The tangent line made by point B (a, 0), and the intersection point of the tangent line and X axis is a (x, 0)
There are: X / r = R / a.x = R & amp; sup2 / A.A (R & amp; sup2 / A, 0)
Any point C (a, b) on a straight line leads to two tangents of a circle
The equation of circle with OC as diameter is: (x-a / 2) & amp; sup2; + (y-b / 2) & amp; sup2; = (A & amp; sup2; + B & amp; sup2;) / 4
The simultaneous solution of two circular equations gives R & amp; sup2; - A (x + y) = 0
This is the equation of the line between the two tangent points, obviously passing through point a

It is proved that the line between the tangent point and the center of the circle is perpendicular to the tangent line
To the contrary
(except in this way)
The tangent passes through the tangent point, and the distance from the tangent point to the center of the circle is the radius of the circle
If the line between the tangent point and the center of the circle is not perpendicular to the tangent, then the vertical line from the tangent point to the center of the circle must be greater than the distance from the tangent to the center of the circle, that is to say, there must be a second intersection between the tangent and the circle, which does not conform to the definition of tangent
Therefore, the hypothesis does not hold, so the two must be vertical
It doesn't matter. I don't mind if I can work out the concrete composition, calculus or Einstein's mass energy equation!!! But don't talk big about theory or special value method...

The way is to know the slope of the tangent Then the slope of the straight line connecting the tangent point and the dot is calculated If the distance between the dot and the line is equal to the distance between the dot and the tangent point, it will be vertical (and the tangent slope must be known)! So if we don't know the tangent slope, we can't prove I don't know if your proof is correct Ask the teacher Sometimes it's misleading and hard to understand!
Add your: I said the two methods know that the slope can be proved
Let the coordinates of the center of the circle be (a, b), the tangent point be (P, s), the tangent slope be K, and the slope of the line from the center of the circle to the tangent point be K1
The first kind
S-B / P-A = K1, if you know the slope k, you can see that they can not multiply by - 1! If they multiply by - 1, they will be vertical. Do you understand (calculate the concept of multiplying the vertical slope of two straight lines by - 1, Y-Y1 / x-x1 = the slope of the line between two points)?
The second kind
The tangent equation is y-p = K (X-S) kx-y-ks + P = 0
The distance from the center of the circle to the tangent is:
K & sup2; + 1 under "ak-b-ks + P" / radical (distance from formula point to straight line)
The distance between the center of the circle and the tangent point is
(A-P) & sup2; + (B-S) & sup2;
If you know the tangent slope, you can substitute it into the slope k to see whether the two distances are equal or not. (if they are equal, they must be vertical, because the distance from the center of the circle to the tangent is the distance perpendicular to the tangent. If the distance from the center of the circle to the tangent is equal to the distance, they are a straight line and must be perpendicular to the tangent.)
Maybe there's something else I haven't learned!