45°,线段乘积相等

hbghlyj

AO=BO,AG=BG,C在AB上,D为BC中点,AO⊥CO,AH⊥CH,E在BO上,∠ADE=45°,F在DE上,∠DBF=∠BED,BG交CH于I,求证BO⋅BI=2BE⋅BF.

hbghlyj

上来就把所有能连的都连上
丢给计算机去找结论

1. GeometricScene[{b -> {0.2482769139, 1.4686430004},
2.   a -> {-3.280223225, 1.5093223184},
3.   c -> {-0.8811841203, 1.4816643133},
4.   d -> {-0.3164536032, 1.4751536569},
5.   e -> {-1.1532216112, 2.3314405299},
6.   f -> {-0.5026518145, 1.6656952199},
7.   g -> {-1.510885681, 1.9302672274},
8.   h -> {-1.0159271708, 2.048023314},
9.   i -> {-0.9530880874, 1.7838948092},
10.   o -> {-1.5037726368, 2.547248505},
11.   j -> {-1.5159731558, 1.4889826594}}, {Line[{a, o}], Line[{b, e, o}],
12.    Line[{a, j, c, d, b}], Line[{c, o}], Line[{g, i, f, b}],
13.   Line[{h, i, c}], Line[{a, g, h}], Line[{e, f, d}], Line[{o, h}],
14.   Line[{c, e}], Line[{c, f}], Line[{d, h}], Line[{d, i}],
15.   Line[{f, h}], Line[{d, g}], Line[{e, h}], Line[{e, i}],
16.   Line[{e, g}], Line[{e, a}], Line[{o, g, j}], Line[{e, j}],
17.   Line[{h, j}], Line[{i, j}], Line[{f, j}], Line[{o, i}],
18.   Line[{o, f}], Line[{a, i}], Line[{a, f}], Line[{b, h}]}]

Copy the Code

hbghlyj

j是ab中点.
GeometricAssertion[{Triangle[{o,g,a}],Triangle[{o,g,b}]},"Congruent"],
GeometricAssertion[{Triangle[{j,a,o}],Triangle[{j,b,o}]},"Congruent"],
GeometricAssertion[{Triangle[{j,a,g}],Triangle[{j,b,g}]},"Congruent"],
GeometricAssertion[{Triangle[{o,a,c}],Triangle[{j,a,o}],Triangle[{j,b,o}],Triangle[{j,o,c}]},"Similar"],
GeometricAssertion[{Triangle[{o,g,a}],Triangle[{o,g,b}],Triangle[{h,o,a}]},"Similar"],
GeometricAssertion[{Triangle[{h,a,c}],Triangle[{j,a,g}],Triangle[{j,b,g}]},"Similar"],
GeometricAssertion[{Triangle[{e,b,f}],Triangle[{o,c,h}]},"Similar"],
GeometricAssertion[{Triangle[{d,e,c}],Triangle[{d,c,f}]},"Similar"],
GeometricAssertion[{Triangle[{d,e,b}],Triangle[{d,b,f}]},"Similar"],d==Midpoint[{c,b}],
j==Midpoint[{a,b}],EuclideanDistance[a,o]==EuclideanDistance[o,b],
EuclideanDistance[a,g]==EuclideanDistance[g,b],
EuclideanDistance[e,d]==EuclideanDistance[c,f]+EuclideanDistance[f,b],
PlanarAngle[{d,b,e}]==PlanarAngle[{d,f,b}]==
PlanarAngle[{e,f,i}]==PlanarAngle[{g,o,c}]==PlanarAngle[{j,a,o}],
PlanarAngle[{b,e,f}]==PlanarAngle[{d,b,f}]==PlanarAngle[{e,o,h}]==PlanarAngle[{h,o,c}]==PlanarAngle[{j,a,g}],
PlanarAngle[{a,g,o}]==PlanarAngle[{a,o,h}]==PlanarAngle[{d,c,i}]==PlanarAngle[{h,g,j}]==PlanarAngle[{i,g,o}],
PlanarAngle[{a,o,g}]==PlanarAngle[{e,o,g}]==PlanarAngle[{g,h,o}]==PlanarAngle[{j,c,o}],
PlanarAngle[{a,g,j}]==PlanarAngle[{h,g,o}]==PlanarAngle[{i,g,j}]==PlanarAngle[{j,c,i}],
PlanarAngle[{e,b,f}]==PlanarAngle[{g,a,o}]==PlanarAngle[{i,c,o}],
PlanarAngle[{d,f,i}]==PlanarAngle[{e,f,b}]==PlanarAngle[{o,h,i}],
PlanarAngle[{a,o,e}]==PlanarAngle[{f,i,h}]==PlanarAngle[{g,i,c}],
PlanarAngle[{c,d,f}]==PlanarAngle[{h,o,g}]==45*Degree,
PlanarAngle[{f,i,c}]==PlanarAngle[{g,i,h}],
PlanarAngle[{f,e,h}]==PlanarAngle[{o,h,e}],
PlanarAngle[{e,o,c}]==PlanarAngle[{h,g,i}],
PlanarAngle[{e,f,o}]==PlanarAngle[{h,o,f}],PlanarAngle[{d,f,h}]==PlanarAngle[{o,h,f}],
PlanarAngle[{c,f,e}]==PlanarAngle[{j,c,e}],
PlanarAngle[{c,f,d}]==PlanarAngle[{d,c,e}],
PlanarAngle[{c,f,b}]==PlanarAngle[{o,e,c}],
PlanarAngle[{c,e,f}]==PlanarAngle[{d,c,f}],
PlanarAngle[{b,e,c}]==PlanarAngle[{c,f,i}],
PlanarAngle[{b,d,f}]==135*Degree,
EuclideanDistance[e,b]==EuclideanDistance[o,c]+EuclideanDistance[o,e],
EuclideanDistance[a,o]==EuclideanDistance[e,b]+EuclideanDistance[o,e],
EuclideanDistance[a,d]==EuclideanDistance[j,b]+EuclideanDistance[j,d],
EuclideanDistance[a,c]==EuclideanDistance[j,b]+EuclideanDistance[j,c],
EuclideanDistance[a,j]==EuclideanDistance[c,b]+EuclideanDistance[j,c],
EuclideanDistance[f,b]==EuclideanDistance[i,c]+EuclideanDistance[i,f],
EuclideanDistance[a,g]==EuclideanDistance[g,i]+EuclideanDistance[i,b],
EuclideanDistance[a,h]==EuclideanDistance[g,b]+EuclideanDistance[g,h],
EuclideanDistance[a,g]==EuclideanDistance[f,b]+EuclideanDistance[g,f],
EuclideanDistance[a,j]==EuclideanDistance[d,b]+EuclideanDistance[j,d],
EuclideanDistance[j,d]==EuclideanDistance[d,b]+EuclideanDistance[j,c],
EuclideanDistance[a,d]==EuclideanDistance[a,c]+EuclideanDistance[d,b],
EuclideanDistance[j,b]==EuclideanDistance[c,d]+EuclideanDistance[j,d],
EuclideanDistance[a,j]==EuclideanDistance[c,d]+EuclideanDistance[j,d],
EuclideanDistance[a,b]==EuclideanDistance[a,d]+EuclideanDistance[c,d],GeometricAssertion[{Line[{a,j,c,d,b}],Line[{j,g,o}]},
"Perpendicular"],GeometricAssertion[{Line[{a,g,h}],Line[{h,i,c}]},"Perpendicular"],
GeometricAssertion[{Line[{a,o}],Line[{o,c}]},"Perpendicular"],PlanarAngle[{a,j,g}]==PlanarAngle[{a,o,c}]==PlanarAngle[{c,j,g}]==PlanarAngle[{g,h,i}]==90*Degree,
GeometricAssertion[{Line[{e,f,d}],Line[{o,h}]},"Parallel"]}]

hbghlyj

回复 1# hbghlyj
想不清楚
OH平分COB

hbghlyj

继续加辅助元素
设C关于OH对称到U,BU中点为V,新产生的结论为
OE=UV=VB
CU∥DV
BG∥HU
PlanarAngle[{a, u, c}] == PlanarAngle[{g, o, i}],
PlanarAngle[{a, u, e}] == PlanarAngle[{g, i, o}],
PlanarAngle[{a, u, h}] == PlanarAngle[{e, o, i}],
PlanarAngle[{a, u, v}] == PlanarAngle[{f, i, o}],
PlanarAngle[{b, f, u}] == PlanarAngle[{h, u, f}],
PlanarAngle[{c, f, i}] == PlanarAngle[{u, e, c}],
PlanarAngle[{c, o, i}] == PlanarAngle[{g, a, u}],
PlanarAngle[{c, u, d}] == PlanarAngle[{v, d, u}],
PlanarAngle[{d, v, c}] == PlanarAngle[{u, c, v}],
PlanarAngle[{f, i, u}] == PlanarAngle[{h, u, i}],
PlanarAngle[{h, i, o}] == PlanarAngle[{o, a, u}],
PlanarAngle[{h, o, i}] == PlanarAngle[{j, a, u}],
PlanarAngle[{h, u, g}] == PlanarAngle[{i, g, u}],
PlanarAngle[{i, f, h}] == PlanarAngle[{u, h, f}],
PlanarAngle[{u, h, v}] == PlanarAngle[{v, b, h}],
PlanarAngle[{f, b, h}] == PlanarAngle[{u, h, b}] ==
PlanarAngle[{u, v, h}],
PlanarAngle[{f, i, c}] == PlanarAngle[{g, i, h}] ==
PlanarAngle[{u, h, i}],
PlanarAngle[{b, d, f}] == PlanarAngle[{c, d, v}] ==
PlanarAngle[{j, c, u}] == 135 \[Degree],
PlanarAngle[{d, f, i}] == PlanarAngle[{e, f, b}] ==
PlanarAngle[{o, h, i}] == PlanarAngle[{o, h, u}],
PlanarAngle[{e, u, h}] == PlanarAngle[{g, a, o}] ==
PlanarAngle[{i, c, o}] == PlanarAngle[{v, b, f}],
PlanarAngle[{b, d, v}] == PlanarAngle[{c, d, f}] ==
PlanarAngle[{d, c, u}] == PlanarAngle[{h, o, g}] == 45 \[Degree],
PlanarAngle[{d, b, f}] == PlanarAngle[{e, o, h}] ==
PlanarAngle[{h, o, c}] == PlanarAngle[{j, a, g}] ==
PlanarAngle[{u, e, f}],
PlanarAngle[{d, b, v}] == PlanarAngle[{d, f, b}] ==
PlanarAngle[{e, f, i}] == PlanarAngle[{g, o, c}] ==
PlanarAngle[{j, a, o}],
PlanarAngle[{a, j, g}] == PlanarAngle[{a, o, c}] ==
PlanarAngle[{c, j, g}] == PlanarAngle[{g, h, i}] ==
PlanarAngle[{v, d, f}] == 90 \[Degree],
PlanarAngle[{a, o, g}] == PlanarAngle[{c, u, h}] ==
PlanarAngle[{e, o, g}] == PlanarAngle[{g, h, o}] ==
PlanarAngle[{j, c, o}] == PlanarAngle[{u, c, i}],
PlanarAngle[{a, g, j}] == PlanarAngle[{c, u, e}] ==
PlanarAngle[{d, v, u}] == PlanarAngle[{h, g, o}] ==
PlanarAngle[{i, g, j}] == PlanarAngle[{j, c, i}] ==
PlanarAngle[{u, c, o}],
PlanarAngle[{a, g, o}] == PlanarAngle[{a, o, h}] ==
PlanarAngle[{c, u, v}] == PlanarAngle[{d, c, i}] ==
PlanarAngle[{d, v, b}] == PlanarAngle[{h, g, j}] ==
PlanarAngle[{i, g, o}]
GeometricAssertion[{Triangle[{h, b, u}],
  Triangle[{v, h, u}]}, "Similar"]
GeometricAssertion[{Triangle[{h, a, c}], Triangle[{j, a, g}],
   Triangle[{j, b, g}], Triangle[{d, e, v}]}, "Similar"]