# Take any quadrilateral. Let A, B, C and D be its vertices. Let P, Q, R and S be the middle points of its edges. Show that the quadrilateral PQRS is a parallelogram. Hint: Show that vector SP=...

Take any quadrilateral. Let A, B, C and D be its vertices. Let P, Q, R and S be the middle points of its edges.

Show that the quadrilateral PQRS is a parallelogram.

Hint: Show that vector SP= vector RQ and that vector SR= vector PQ

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### 1 Answer

You need to use the midline theorem to prove that `PQRS` is a parallelogram.

You need to start by creating the following triangles `Delta ABC` and `Delta ADC` (`AC` represents the diagonal of quadrilateral). Notice that the triangles `Delta ABC` and `Delta ADC` have the common base AC.

The problem provides the information that ` P,Q` are the midpoints of the sides `AB` and `BC` , hence, the line `PQ` represents the midline of triangle `Delta ABC` and `PQ` is a segment parallel to the base `AC` and it is half as long.

The problem provides the information that `R,S` are the midpoints of the sides `CD` and `DA` , hence, the line `RS` represents the midline of triangle `Delta ADC` and `RS` is a segment parallel to the base `AC` and it is half as long.

Since `{(PQ||AC,PQ = (AC)/2),(RS||AC,RS=(AC)/2):} => PQ||RS; PQ=RS` .

Reasoning by analogy, you may use the next diagonal BD to create the triangle `Delta BCD` and `Delta DAB` and to prove that `QR = PS` and `QR||PS.`

**Since the geometric shape PQRS has the opposite sides parallel and equal, `QR = PS;QR||PS` and `PQ||RS; PQ=RS` yields that `PQRS ` represents a parallelogram.**