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Vignettes of Ancient Mathematics
Hero of Alexandria, Metrica, Prop. 8: a general procedure for finding areas of triangles without finding the altitude.
%7BE%7D;%0D%0A%5Cdraw%20(-1.3,5.8)node%20%5Babove%5D%20%20%7BA%7D--(-2.2,-1.2)edge(-2.2,-5.9)edge%20(-6.6,-1.2)node%20%5Bbelow%20left%5D%20%20%7BB%7D--(6.6,-1.2)edge(-2.2,-5.9)node%20%5Bbelow%5D%20%7B%24%5CGamma%24%7D--cycle(-1.3,5.8)%20--%20(0.45,1.143)edge(-2.2,-5.9)edge(2,2.9)edge(0.45,-1.2)edge%20(-1.85,1.4)edge(6.6,-1.2)circle(2.335)--(-2.2,-1.2)%20;%0D%0A%5Cnode%20at(-6.6,-1.6)%7B%24%5CTheta%24%7D;%0D%0A%5Cnode%20at(-2.2,-6.3)%7B%24%5CLambda%24%7D;%0D%0A%5Cnode%20at%20(0.6,1.8)%20%7BH%7D;%0D%0A%5Cnode%20at%20(-2.3,1.4)%20%7B%24%5CDelta%24%7D;%0D%0A%5Cnode%20at%20(-0.3,-1.6)%20%7BK%7D;%0D%0A%5Cnode%20at%20(2.2,3.15)%20%7BZ%7D;%0D%0A%5Cend%7Btikzpicture%7D) 三角形ABΓ的内切圆H在各边上的切点为 E, Z, Δ.
延长ΓB到Θ使ΒΘ = ΑΔ, 则2ΘΓ=AB+BΓ+ΓA, 则ABΓ的面积等于ΗΕ · ΘΓ.
过H作HΓ垂线, 过B作BΓ垂线, 交于Λ.
HΛ与BΓ交于K.
因为BΓΛH共圆, 所以∠BΛΓ=180°-∠BHΓ=∠AHΔ, 所以直角三角形 ΑΗΔ 与 ΓΒΛ 相似. 剩下的就是导比例:
ΒΓ : ΒΛ = ΑΔ : ΔΗ (因为 ΑΗΔ 与 ΓΒΛ 相似)
= ΒΘ : ΕΗ (因为 ΒΘ = ΑΔ 和 ΕΗ = ΔΗ)
ΒΓ : ΒΘ = ΒΛ : ΕΗ (移项)
= ΒΚ : ΚΕ (因为 ΕΚΗ 与 ΒΚΛ 相似)
ΘΓ : ΒΘ = ΒΕ : ΚΕ (两边分数加1)
ΘΓ² : ΒΘ·ΘΓ = ΒΕ·ΕΓ : ΓΕ·ΕΚ (分子分母同乘ΘΓ和ΓΕ)
代入 ΕΗ² = ΓΕ·ΕΚ (因为直角三角形 ΚΕΗ ∼ ΚΗΓ ∼ ΗΕΓ)得
ΘΓ² : ΒΘ·ΘΓ = ΒΕ·ΕΓ : ΕΗ²
ΘΓ² · ΕΗ² = ΒΘ·ΘΓ · ΒΕ·ΕΓ (变为乘积式)
$\rm ΑΒΓ的面积 = \sqrt{ΗΕ^2 · ΘΓ^2} = \sqrt{ΒΘ·ΘΓ·ΒΕ·ΕΓ}$
ΒΘ = 1/2 perimeter - ΓΒ
ΘΓ = 1/2 perimeter
ΒΕ = 1/2 perimeter - ΑΓ
ΕΓ = 1/2 perimeter - ΑΒ
海伦公式证毕. |
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