It is proved that the midpoint quadrilateral in the diamond judgment method: the quadrilateral obtained by connecting the midpoints of the sides of the quadrilateral in turn is called the midpoint quadrilateral. No matter how the shape of the original quadrangle changes, the shape of the midpoint quadrangle is always a parallelogram.
The midpoint quadrangle of a rhombus is a rectangle (the midpoint quadrangle of a quadrangle with vertical diagonal lines is a rhombus, and the midpoint quadrangle of a quadrangle with equal diagonal lines is a rectangle). )
A diamond is defined on the premise of a parallelogram. Firstly, it is a parallelogram, but it is a special parallelogram, which is characterized by "a group of adjacent sides are equal", thus adding some special properties and judgment methods different from parallelogram. Calculation of diamond area: 1. Half the diagonal product. (Any quadrilateral with diagonal lines perpendicular to each other will do); By dividing the diamond into two triangles and simplifying them; 2. The bottom is multiplied by the height; 3. Let the side length of the diamond be a and the included angle be θ, then the area formula is: s = a 2 sin θ.
A set of parallelograms with equal adjacent sides is a diamond.
2. A quadrilateral with four equilateral sides is a diamond.
3. Parallelograms with diagonal lines perpendicular to each other are rhombic.
Prove the diamond decision theorem;
AB = CD，BC=AD，
∴ quadrilateral ABCD is a parallelogram.
AB = BC，
∴ Quadrilateral ABCD is a diamond (a group of parallelograms with equal adjacent sides is a diamond).
2. Parallelograms with diagonal lines perpendicular to each other are diamonds.
Prove:
∵ quadrilateral ABCD is a parallelogram,
∴ OA=OC (diagonal bisection of parallelogram).
∵AC⊥BD，
∴ BD's straight line is the middle vertical line of segment AC,
Ab = BC,
∴ Quadrilateral ABCD is a diamond (a group of parallelograms with equal adjacent sides is a diamond).
3. A set of parallelograms with equal adjacent sides is a diamond.
RF is the center line of triangle ABD, so RF∑AD,
Similarly: GH∨AD, RH∨BE, FG∨BE, so there are RF∨GH, RH∨FG,
So the quadrilateral RFGH is a parallelogram;
In the second step, it is proved that △ ACD △ BCE, then AD=BE, so there is RH = RF, so the quadrilateral RFGH is a diamond.
Prove the definition of known diamond judgment: As shown in the figure, in◇▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ρ▽▽96 The quadrilateral AFCE is a diamond.
Prove:
∵ quadrilateral ABCD is a parallelogram,
∴AE∨fc (the opposite sides of a parallelogram are parallel),
∴ ∠EAO=∠FCO。
∫EF divides AC equally,
∴ AO=OC。
∠∠AOE =∠COF = 90 degrees,
∴△AOE?△cof(asa)，
∴ EO=FO，
∴ Quadrilateral AFCE is a parallelogram (a quadrilateral with its diagonal bisected is a parallelogram).
And ∵EF⊥AC,
∴ Quadrilateral AFCE is a diamond (parallelograms with diagonal lines perpendicular to each other are diamonds).
It is proved that three examples of diamond judgment method 1 and quadrilateral with equal sides are diamonds.
Prove:
AB = CD，BC=AD，
∴ Quadrilateral ABCD is a flat-bladed quadrilateral (two groups of quadrangles with equal opposite sides are parallelograms).
AB = BC，
∴ Quadrilateral ABCD is a diamond (a group of parallelograms with equal adjacent sides is a diamond).
2. Parallelograms with diagonal lines perpendicular to each other are diamonds.
Prove:
∵ quadrilateral ABCD is a parallelogram,
∴OA=OC (the diagonal of the parallelogram is equally divided).
∵AC⊥BD，
∴BD's straight line is the middle vertical line of segment AC,
∴AB=BC，
∴ Quadrilateral ABCD is a diamond (a group of parallelograms with equal adjacent sides is a diamond).
3. A set of parallelograms with equal adjacent sides is a diamond.
RF is the center line of triangle ABD, so RF∑AD,
Similarly: GH∨AD, RH∨BE, FG∨BE, so there are RF∨GH, RH∨FG,
So the quadrilateral RFGH is a parallelogram;
In the second step, it is proved that △ ACD △ BCE, then AD=BE, so there is RH = RF, so the quadrilateral RFGH is a diamond.
Articles proving the diamond judgment method:
★ Similar knowledge points of special parallelogram graphics in junior high school mathematics.
★ Summary of Junior High School Mathematics Knowledge Points (Shanghai Science Edition)
★ Mathematics teaching methods and skills
★ Three-year knowledge points induction of junior high school mathematics
★ People's Education Edition Diamond Teaching Plan
★ Knowledge points of mathematics outline for senior high school entrance examination
★ Inductive analysis and problem-solving steps of senior one mathematics.
★ Judgment and reflection on parallelogram mathematics teaching plan
★ Mid-term examination paper of eighth grade mathematics
★ Prove parallelogram method
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