if the diagonals of quadrilateral ABCD intersect at their midpoints, then the quadrilateral is a parallelogram.

Question

if the diagonals of quadrilateral ABCD intersect at their midpoints, then the quadrilateral is a parallelogram.

Prove that AM MC
--- = ---
MB MD

Ive tried figuring out this question for a while now and I cant figure it out.

Answer 1

*******Prove that AM/BM = MC/MD

Answer 2

If AC and BD intersect at their mid-points, AM=MC, and BM=MD.

Therefore
AM/MC=BM/MD=1
Transpose to get:
AM/BM=MC/MD

Answer 3

To prove that the diagonals of quadrilateral ABCD intersect at their midpoints, and thus have the property that (AM / MB) = (MC / MD), we can use the properties of triangles and proportionality.

Let's start by assuming that the diagonals of quadrilateral ABCD intersect at points P and Q, and that they intersect at their midpoints. In other words, AP = PB and CQ = QD.

Now, we can see that triangles APQ and CPQ share an angle at point Q. By the Angle-Angle Similarity Theorem, we can conclude that triangles APQ and CPQ are similar.

In similar triangles, the ratio of the lengths of corresponding sides is equal. So, we have:

(AP / CP) = (AQ / CQ)

Since AP = PB and CQ = QD, we can substitute these values:

(PB / CP) = (AQ / QD)

Now, let's consider triangles ABP and CDQ. These triangles also share an angle at point Q. By the Angle-Angle Similarity Theorem, we can deduce that triangles ABP and CDQ are similar.

Again, in similar triangles, the ratio of the lengths of corresponding sides is equal. So, we have:

(AB / CD) = (BP / DQ)

Now, let's use these two ratios to find a relation between the segments AM and MC, and MB and MD.

Since AB = CD (opposite sides of a parallelogram are equal), we can substitute that:

(PB / CP) = (AQ / QD) = (BP / DQ)

Now, let's manipulate the ratio:

(PB / CP) = (BP / DQ)

Cross-multiply to get:

PB * DQ = CP * BP

Rearranging the terms:

PB * DQ - BP * DQ = CP * BP - BP * DQ

Distributing the terms:

DQ(PB - BP) = BP(CP - DQ)

Simplifying further:

DQ(0) = BP(CP - DQ)

Since PB = BP, we can remove it from both sides:

DQ(0) = CP - DQ

Simplifying:

0 = CP - DQ

Now, let's rearrange this equation to find the relation between AM, MC, MB, and MD:

CP = DQ

Therefore, AM / MB = MC / MD, since CP = DQ.

This proves that if the diagonals of quadrilateral ABCD intersect at their midpoints, then the quadrilateral is a parallelogram and has the property that (AM / MB) = (MC / MD).