Visitor

2016-06-02 09:28:27

#1

Let ABCD be a quadrilateral some. Demonstrate that its sides means are the peaks of a parallelogram: a) in the case of convex quadrilateral ABCD. b) if the concave quadrilateral ABCD.

Visitor

2016-06-02 10:51:22

#2

We have the figure below, with convex quadrilateral ABCD and M, N, P and Q means its sides.

In the figure above we have:

1) M - mid AB

2) N - mid BC

From these two relationships => MN - medium line in ?ABC => MN || MN = AC and AC / 2

In analogy to prove that PQ - middle line in the triangle ADC => PQ || AC and AC = PQ / 2

So we have MN || AC || PQ => MN || PQ

and MN = PQ = AC / 2

As we know, a convex quadrilateral has two opposite sides parallel and congruent is parallelogram => MNPQ - parallelogram

In the figure above we have:

1) M - mid AB

2) N - mid BC

From these two relationships => MN - medium line in ?ABC => MN || MN = AC and AC / 2

In analogy to prove that PQ - middle line in the triangle ADC => PQ || AC and AC = PQ / 2

So we have MN || AC || PQ => MN || PQ

and MN = PQ = AC / 2

As we know, a convex quadrilateral has two opposite sides parallel and congruent is parallelogram => MNPQ - parallelogram

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