如何通过三角形中线中点计算面积？附坐标实例求解

Hey there! Let's tackle this problem in two parts—first breaking down the method, then solving the specific example you provided.

First, let's clarify the key relationship between the "median-midpoint triangle" (the triangle formed by connecting the midpoints of a triangle's medians) and the original triangle:

• The original triangle’s sides are 4 times longer than the corresponding sides of the median-midpoint triangle, and they share identical angles (they’re similar triangles).
• Since the area of similar triangles scales with the square of their side lengths, the original triangle’s area will be (4^2 = 16) times the area of the median-midpoint triangle.

Here’s the step-by-step method:

• Calculate the area of the triangle formed by the three given median midpoints (let’s call this (S_{\text{mid}})).
• Multiply that area by 16 to get the original triangle’s area: (S_{\text{original}} = 16 \times S_{\text{mid}}).

If you prefer a coordinate-based approach, you can solve for the original triangle’s vertices using the median midpoints (P(p_1,p_2)), (Q(q_1,q_2)), (R(r_1,r_2)):

• (A(x_1,y_1) = (3p_1 - q_1 - r_1, 3p_2 - q_2 - r_2))
• (B(x_2,y_2) = (3q_1 - p_1 - r_1, 3q_2 - p_2 - r_2))
• (C(x_3,y_3) = (3r_1 - p_1 - q_1, 3r_2 - p_2 - q_2))
  Then use the shoelace formula to compute the original area directly—this will confirm the 16x scaling factor every time.

Let’s apply the method above:

First, calculate the area of the median-midpoint triangle:

• This is a right triangle! The base is the horizontal distance between (1,2) and (8,2): (8-1=7).
• The height is the vertical distance between (1,2) and (1,8): (8-2=6).
• Area (S_{\text{mid}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times7\times6=21).

Now multiply by 16 to get the original triangle’s area:
(S_{\text{original}} =16\times21=336).

To double-check, let’s compute the original vertices using the coordinate formula and verify with the shoelace method:

• Let (P=(1,2)), (Q=(8,2)), (R=(1,8)):
  • (A=(3\times1 -8 -1, 3\times2 -2 -8)=(-6,-4))
  • (B=(3\times8 -1 -1, 3\times2 -2 -8)=(22,-4))
  • (C=(3\times1 -1 -8, 3\times8 -2 -2)=(-6,20))
    Using the shoelace formula on (A(-6,-4)), (B(22,-4)), (C(-6,20)):
    [
    \begin{align*}
    S&=\frac{1}{2}\left| x_1(y_2-y_3) +x_2(y_3-y_1)+x_3(y_1-y_2) \right|\
    &=\frac{1}{2}\left| (-6)(-4-20)+22(20-(-4))+(-6)(-4-(-4)) \right|\
    &=\frac{1}{2}\left| 144+528+0 \right|=\frac{1}{2}\times672=336
    \end{align*}
    ]
    Perfect, this matches our earlier result.

内容的提问来源于stack exchange，提问作者user529689