Try this
Drag any orange dot and note that the red lines always form a parallelogram.

If you find the midpoints of each side of any quadrilateral, then link them sequentially with lines, the result is always a parallelogram. This may seem unintuitive at first, but if you drag any vertex of the quadrilateral above, you will see it is in fact always true, even when the quadrilateral is 'self-crossing' - where some sides of the quadrilateral cross over other sides.

The figure below is the same as above, except with the points J,K,L, M labelled and the line DB added. By definition J,K,L,M are the midpoints of their respective sides.

Argument | Reason | |
---|---|---|

1 | JM is the midsegment of the triangle ABD | The midsegment of a triangle is a line linking the midpoints of two sides (See Midsegment of a triangle) |

2 | JM is half DB and parallel to it | From the properties of the midsegment of a triangle |

3 | Likewise in triangle DBC, LK is also half DB and parallel to it | From the properties of the midsegment of a triangle |

4 | JKLM is a parallelogram | A pair of opposite sides (LK and JM) are parallel and congruent |

- Q.E.D

- Polygon general definition
- Quadrilateral
- Regular polygon
- Irregular polygon
- Convex polygons
- Concave polygons
- Polygon diagonals
- Polygon triangles
- Apothem of a regular polygon
- Polygon center
- Radius of a regular polygon
- Incircle of a regular polygon
- Incenter of a regular polygon
- Circumcircle of a polygon
- Parallelogram inscribed in a quadrilateral

- Square
- Diagonals of a square
- Rectangle
- Diagonals of a rectangle
- Golden rectangle
- Parallelogram
- Rhombus
- Trapezoid
- Trapezoid median
- Kite
- Inscribed (cyclic) quadrilateral

- Regular polygon area
- Irregular polygon area
- Rhombus area
- Kite area
- Rectangle area
- Area of a square
- Trapezoid area
- Parallelogram area

- Perimeter of a polygon (regular and irregular)
- Perimeter of a triangle
- Perimeter of a rectangle
- Perimeter of a square
- Perimeter of a parallelogram
- Perimeter of a rhombus
- Perimeter of a trapezoid
- Perimeter of a kite

- Exterior angles of a polygon
- Interior angles of a polygon
- Relationship of interior/exterior angles
- Polygon central angle

- Tetragon, 4 sides
- Pentagon, 5 sides
- Hexagon, 6 sides
- Heptagon, 7 sides
- Octagon, 8 sides
- Nonagon Enneagon, 9 sides
- Decagon, 10 sides
- Undecagon, 11 sides
- Dodecagon, 12 sides

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