Senior high school olympiad 2021-1-15

202 1- 1 1- 15-0 1

(Source: Mathematical Olympiad Series Second Edition High School Volume Plane Geometry Fan Duan Xi Deng Bowen Selected Lecture on Other Methods and Problems of Plane Geometry P 1 18 Exercise 05)

The point is a point outside the circle, the intersection point is the tangent of the circle, and the tangent points are respectively, and the straight line passing through the point and parallel to the point and the intersection point. Proof: No matter how it changes, it will always cross a certain point.

certificate

The right-hand rectangular coordinate system is established with the origin and straight line as the axis, and the radius can be set to.

, , .

Take the intersection with the axis, and then.

Set a straight line,

So ...

In the same way.

Then. ( 1)

In (1), so

The next certificate: three-point * * line.

(2)

Due to.

Formula (2) is clearly established.

So, the three-point line.

It is also a fixed point, so it is a fixed point.

So it is a fixed point.

So, you can get the certificate by fixing the point.

202 1- 1 1- 15-02

(Source: Mathematical Olympiad Series Second Edition High School Volume Plane Geometry Fan Duan Xi Deng Bowen Selected Lecture on Other Methods and Problems of Plane Geometry P 1 18 Exercise 06)

, are two fixed points on the plane, a moving point on the same side of a straight line on the plane, with, as the side and a square outside. It is proved that no matter which point on the same side of a straight line is taken, the position of the midpoint m of the line remains unchanged.

certificate

Let each letter in the picture represent the complex number of the corresponding point.

So this has nothing to do with.

202 1- 1 1- 15-03

(Source: Mathematical Olympiad Series Second Edition High School Volume Plane Geometry Fan Duan Xi Deng Bowen Selected Lecture on Other Methods and Problems of Plane Geometry P 1 18 Exercise 07)

It is proved that the midpoints of the midpoints of all sides of any convex quadrilateral must coincide.

certificate

A graph is an arbitrary quadrilateral, where,, and are the midpoint of each side.

Because,,,.

Therefore, the midpoint of is.

The midpoint of is.

From this we know.

This is a result to be proved.

202 1- 1 1- 15-04

(Source: Mathematical Olympiad Series Second Edition High School Volume Plane Geometry Fan Duan Xi Deng Bowen Selected Lecture on Other Methods and Problems of Plane Geometry P 1 18 Exercise 08)

Make a regular triangle, a regular triangle, a regular triangle and a regular triangle on the outside of the convex quadrilateral. Remember that the sum of diagonal lines of the quadrilateral is the maximum. (The 7th Women's Mathematical Olympiad in 2008)

solve

If the quadrilateral is a square, you can get it.

The following proofs:

Let, and be the midpoints of edges and, respectively, while the midpoints of, and are,, and respectively.

It is a parallelogram.

Link and set point are the midpoint of, respectively.

and

So,

Therefore, it is a regular triangle.

Similarly, it is also a regular triangle.

Let be the midpoint of the difference, so there is

Similarly,