Geometric Verification If the midpoints of each side of a quadrilateral form a square, then the quad

The midpoint of the four sides of any quadrilateral with equal and perpendicular diagonals can form a square, but the original diagonal does not have the condition of bisecting each other, so the original quadrilateral is not necessarily a square.

So this proposition is a false proposition, and whoever can prove it can only say that he is a master of paradoxes, hehe.

Connect the two diagonal lines of the original quadrilateral;

It forms two triangles with the adjacent sides;

It can be seen that the two adjacent sides of the square that connect the midpoints of each side are their median lines.

So "the two median lines are perpendicular and equal".

then the two diagonals of the original quadrilateral are equal and bisected perpendicular to each other.

The quadrilateral is also a square.

Proof: It seems that as long as the diagonals are vertical and equal, they don't have to be square.

The original quadrilateral is not necessarily a square.

Summary. Yes, the diagonal areas of the four quadrilaterals formed by the line from any point in the square to the midpoint of each side are equal. First, connect the adjacent vertices of the square to get four edges, which are bisected from each other.

Therefore, the midpoint of each edge can be thought of as the intersection of the perpendicular lines that are extended to each side by two vertices. In this way, the diagonal of each quadrilateral is the diagonal of the square, and the diagonal lines are equal in length. Next, we can divide the square into four small squares.

For each small square, the length of its diagonal is equal to the length of the sides of the square, so its diagonal length is equal to that of other small squares. Since the diagonals of the small squares are perpendicular and bisected from each other, they are similar. Finally, consider the diagonal length and area of each quad.

Since the diagonal lengths are equal and each quadrilateral is similar, they have the same diagonal and area proportions. Therefore, the product of the diagonal length and area of each quadrilateral is equal, which means that the diagonal area sum of the four quadrilaterals is equal.

Yes, the diagonal areas of the four four-year-old Zheng sides formed by the line from any point in the square to the midpoint of each side are equal. First, connect the adjacent vertices of the square to get four edges, which are bisected from each other. Therefore, the midpoint of each edge can be thought of as the intersection of the perpendicular lines that are extended to each side by two vertices.

In this way, the diagonal of each quadrilateral is the diagonal of the square, and the diagonal lines are equal in length. Next, we can divide the square into four small squares. For each small square, the length of its diagonal is equal to the length of the sides of the square, so its diagonal length is equal to that of other small squares.

Since the diagonal vertical of the small squares is straight and bisected from each other, they are similar. Finally, consider the diagonal length and area of each quad. Since the diagonal lengths are equal and each quadrilateral is similar, they have the same diagonal and area proportions.

Therefore, the product of the diagonal length and area of each quadrilateral is equal, which means that the area and area of the opposite corners of the four quadrilaterals are equal.

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It can be proved by plural methods.

Only the remainder of the forest group needs to prove that each adjacent side of the midpoint quadrilateral obtained by this rolling is vertically equal.

For two known squares, you may wish to let two of them or the edges of the square be a, a*i; b,b*i

Then the critical edges of the resulting quadrilateral are (a+b) 2 , (ai+bi) 2 =(a+b)i 2

It can be seen that the modulus of the two complex numbers is equal, and the radial angle is different, i

That is, the two edges are vertically equal, and the proof is completed.

It can be proved by plural methods.

It is only necessary to prove that each adjacent side of the quadrilateral forest cogram obtained by this roll is vertically equal on each adjacent side.

For two known squares, you may wish to let the two sets of critical edges be a, a*i; b,b*i

Then the critical edges of the resulting quadrilateral are (a+b) 2 , (ai+bi) 2 =(a+b)i 2

It can be seen that the modulus of the two complex numbers is equal, and the radial angle is different, i

That is, the two edges or the two sides are vertically equal, and the proof is completed.

d Make the diagonal of the original quadrilateral first, according to the median line theorem, the opposite sides of the quadrilateral obtained by connecting the midpoints of each side of the quadrilateral in order must be equal and parallel to each other, and the adjacent sides of the square are equal and perpendicular, so the diagonals of the original quadrilateral are perpendicular and equal.

Pay attention to the observation of the question、、A total of two questions, both must be answered。

Why?? d

Square ABCD, upper midpoint E, lower midpoint F, left midpoint G, right midpoint H connect EF, GH, intersect O, connect EG, EH, FG, FHProve that the quadrilateral EGFH is square.

Because oe=of=og=oh

So eg=eh=fg=fh

Let oe=1, then eg=eh=root number 2

So the angle geh = 90 degrees.

The quadrilateral EGFH is a square.

Solution: The new quadrilateral is a square.

Its sides are equal, and the angles are 90°

Each side of the new quadrilateral is parallel to the diagonal of the original quadrilateral and equal to half of the diagonal of the original quadrilateral, and the diagonal of the original quadrilateral should be equal.

The sides of the new quadrilateral are perpendicular.

The diagonal of the original quadrilateral should also be vertical.

The diagonal lines of the original quadrilateral are perpendicular to each other and equal.

Knowledge points: the quadrilateral knowledge obtained by sequentially connecting the midpoints of each side (for easy to say, it is called here"Midpoint quadrilateral") is only related to the diagonal of the original quadrilateral.

If the diagonals of the original quadrilateral are equal, the midpoint quadrilateral is a diamond;

If the diagonals of the original quadrilateral are perpendicular to each other, the midpoint quadrilateral is rectangular;

If the diagonals of the original quadrilateral are perpendicular to each other and equal, then the midpoint quadrilateral is a square.

Therefore: 1) the diagonal of the parallelogram does not have the above special relationship, so the four sides of the point are not shaped or parallelograms;

2) The diagonal of the right-angled trapezoid does not have a special relationship above, so the midpoint quadrilateral is still a parallelogram;

3) The diagonals of the isosceles trapezoidal are equal, so the point quadrilateral is a rhomboid.