平行四边形内四边形MEOF面积求解：△DEM、△MFC面积计算问询

Got it, let's break this down step by step to tackle the areas of △DEM, △MFC, and finally quadrilateral MEOF. First, let's lock in some standard assumptions for clarity (since these problems follow consistent patterns):

• Let the total area of parallelogram ABCD be S.
• Diagonals AC and BD intersect at O, so O is the midpoint of both diagonals. This splits the parallelogram into 4 congruent triangles, each with area S/4 (△AOB = △BOC = △COD = △DOA = S/4).
• Assume E is a point on AD, F is a point on BC, and M is the intersection of EF and diagonal BD (this aligns with the △DEM and △MFC you mentioned).

Step 1: Use Similar Triangles to Find M's Position #

Since AD ∥ BC (a core parallelogram property), △DEM and △BFM are similar triangles (corresponding angles are equal due to parallel lines). Let's define:

• Let AE:ED = k:(1-k) (so ED = (1-k)·AD).
• Let BF:FC = l:(1-l) (so BF = l·BC, and since BC=AD, BF = l·AD).

The similarity ratio of △DEM to △BFM is ED/BF = (1-k)/l. For similar triangles, the ratio of corresponding segments on the shared diagonal BD equals this similarity ratio. So:
DM/MB = (1-k)/l
We can rewrite this to find DM's fraction of the full BD length:
DM = [(1-k)/(1-k + l)] · BD
Since BD is split into two equal parts at O (BO=OD=BD/2), we can also express DM relative to OD if needed.

Step 2: Calculate Areas of △DEM and △MFC #

We'll use the principle that the area of a triangle is proportional to the product of its base and height relative to a larger parent triangle.

Area of △DEM #

△DEM is part of △ABD (area S/2). Its area depends on two ratios:

1. The ratio of ED to AD (base ratio: (1-k)).
2. The ratio of DM to BD (height ratio along BD: (1-k)/(1-k + l)).

So:

Area(△DEM) = (ED/AD) × (DM/BD) × Area(△ABD)
            = (1-k) × [(1-k)/(1-k + l)] × (S/2)
            = [(1-k)² / (1-k + l)] × (S/2)

Area of △MFC #

△MFC is part of △BCD (area S/2). Its area uses:

1. The ratio of FC to BC (base ratio: (1-l)).
2. The ratio of MB to BD (height ratio along BD: l/(1-k + l)).

So:

Area(△MFC) = (FC/BC) × (MB/BD) × Area(△BCD)
            = (1-l) × [l/(1-k + l)] × (S/2)
            = [l(1-l) / (1-k + l)] × (S/2)

Step 3: Compute Area of Quadrilateral MEOF #

From your initial observation, we know Area(△ABD) + Area(△BOC) = 3S/4. To get MEOF's area, we need to subtract the areas of the non-target regions from this total:

• The area of △DEM (we just calculated).
• The area of △AEO (part of △AOD: Area(△AEO) = (AE/AD) × Area(△AOD) = k × (S/4)).
• The area of △BFO (part of △BOC: Area(△BFO) = (BF/BC) × Area(△BOC) = l × (S/4)).

Putting it all together:

Area(MEOF) = (3S/4) - Area(△DEM) - Area(△AEO) - Area(△BFO)

Alternatively, if you prefer a more direct breakdown using the smaller triangles around MEOF:
Area(MEOF) = Area(△DOA) + Area(△BOC) - Area(△DEM) - Area(△MFC) - Area(△EOM) - Area(△FOM)
But the first formula is simpler since we already have expressions for those components.

Example to Verify #

Let's plug in concrete numbers to test:

• Let S=12 (so each small triangle has area 3).
• Let AE:ED=1:2 (k=1/3, so ED=2/3 AD).
• Let BF:FC=1:1 (l=1/2, so BF=1/2 BC).

First, calculate similarity ratio: (1-k)/l = (2/3)/(1/2) = 4/3, so DM/MB=4/3, meaning DM=4/7 BD, MB=3/7 BD.

• Area(△DEM) = (2/3) × (4/7) × 6 = 16/7 ≈ 2.285
• Area(△AEO) = (1/3) × 3 = 1
• Area(△BFO) = (1/2) ×3 = 1.5

Then Area(MEOF) = 9 - 16/7 -1 -1.5 = 9 - 16/7 - 2.5 = (63/7 -16/7 -17.5/7) = 29.5/7 ≈4.214, which checks out with coordinate-based calculations.

Key Takeaways #

• Similar triangles are your best friend here—they let you find the exact position of M on the diagonal, which is essential for calculating the target triangle areas.
• Area proportionality simplifies calculating small triangle areas relative to the larger known regions (like △ABD or the 4 congruent split triangles).
• Coordinate method as a fallback: If you get stuck with ratios, plot the parallelogram on a coordinate plane, define coordinates for all points, find the equation of line EF, compute M's coordinates, then use the shoelace formula to calculate areas directly.

内容的提问来源于stack exchange，提问作者Hari