As shown in the figure, it is known that point O is the intersection of diagonal lines of rectangular ABCD, BE is parallel to AC, and CE is parallel to BD. It is proved that OE and CB are perpendicula

As shown in the figure, it is known that point O is the intersection of diagonal lines of rectangular ABCD, BE is parallel to AC, and CE is parallel to BD. It is proved that OE and CB are perpendicular to each other. Proof: Because? The quadrilateral ABCD is a rectangle (known)

? So what? OA=OB=OC=OD (diagonal lines of a rectangle are equal and equally divided),?

? Because? BE//AC, CE//BD (known),?

? So what? The quadrangle OBEC is a parallelogram (two groups of parallelograms with opposite sides are parallelograms respectively).

Because again? OB=OC (authentication),

? So what? The quadrilateral OBEC is a diamond (a set of parallelograms with equal adjacent sides is a diamond),

? So what? OE is perpendicular to CB (rhombic diagonals are perpendicular to each other).

Related knowledge points:

Several special quadrangles

1) parallelogram? Two sets of quadrilaterals parallel to opposite sides.

Decision theorem one? Two sets of quadrilaterals with equal opposite sides are parallelograms.

2. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

? 3. A quadrilateral whose diagonal is bisected is a parallelogram.

? Four. Two sets of diagonally equal quadrilaterals are parallelograms.

Property theorem 1. The opposite sides of a parallelogram are equal and parallel.

2. The diagonal bisection of parallelogram.

Three. Diagonal angles of parallelogram are equal and adjacent angles are complementary.

2) Rectangular? A parallelogram with right angles is a rectangle.

? Decision theorem 1. A quadrilateral with three right angles is a rectangle.

? 2. Parallelograms with equal diagonals are rectangles.

? Property theorem 1. It has all the characteristics of a parallelogram. ?

Two. ? All four corners of a rectangle are right angles.

3. The diagonals of rectangles are equal.

3) A rhombus has a set of parallelograms with equal adjacent sides.

? Decision theorem 1. A quadrilateral with four equilateral sides is a prism.

2. Parallelograms with diagonal lines perpendicular to each other are diamonds.

A parallelogram whose diagonal bisects a set of diagonal lines is a diamond.

? Property theorem 1. Has all the properties of a parallelogram. ?

All four sides of the diamond are equal.

3. The diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of pairs? Horn.

4) A rectangle with equal adjacent sides is a square. Diamonds with right angles are squares.

A square has all the characteristics of a rectangle and a diamond.

5) Trapezoid A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are trapezoids. The parallel side is called the bottom of the trapezoid, and the non-parallel side is called the waist of the trapezoid.

? An isosceles trapezoid is an isosceles trapezoid.