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# If the midpoints of the sides of a parallelogram taken in succession are joined, the quadrilateral formed is a parallelogram. Synthetic proof and...

If the midpoints of the sides of a parallelogram taken in succession are joined, the quadrilateral formed is a parallelogram. Synthetic proof and prove.

Introduction: Synthetic and analytic techniques can be used to prove and derive many geometric relationships. Additionally, a straight edge and compass can be used to construct a variety of well-known geometric relationships. Task: Theorem : If the consecutive midpoints of the sides of a parallelogram are joined in order, then the quadrilateral formed from the midpoints is a parallelogram. B. Prove the theorem given above in Euclidean geometry using synthetic techniques. 1. Include each step of your proof. 2. Provide written justification for each step of your proof. My answer below. They approved the math as appropriate but kicked back the proof twice. I need improved proofs and revised math as necessary: To start the synthetic process I have drawn a line between point A and point C and another line between point B and point D. ▲ABC One of the two triangles created by adding the line between A and C. BE = EA The midpoint E is equally located between A and B. So, the line segments BE and EA are the same length. BF = FC The midpoint F is equally located between B and C. So, the line segments BF and FC are the same length. ?? ?? = ?? ?? The line segment EF. ±² || °³ Using the triangle proportion property, it shows that EF is parallel to AC.
▲ACD The second triangle from adding the AC line. DH = HA The midpoint H is equally located between A and D. So, the line segments DH and HA are the same length. DG = GC The midpoint G is equally located between C and D. ?? ?? = ?? ?? The line segment HG. So, the line segments DG and GC are the same length. HG || AC By bisecting two side of a triangle, the bisecting line is parallel to the third side; it shows that HG is parallel to AC. HG||AC and EF||AC so, HG||EF Proof that the HG side is parallel to the EF side. ABD One of the two triangles created by adding the line between B and D. BE = EA The midpoint E is equally located between A and B. So, the line segments BE and EA are the same length. AH = HD The midpoint H is equally located between A and D. So, the line segments AH and HD are the same length. ?? ?? = ?? ?? The line segment EH. E H G F A B C D
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Friend! Here is the synthetic proof... View the full answer

Suppose ABCD is a quadrilateral and G, F, E, H are the midpoints of the sides AB, BC, CD,
and DA respectively.
If ABCD has the opposite sites parallel, then it is nothing but a rectangle and thus,...

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