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Senior two mathematics solid geometry
(1) Proof: It is easy to know △A? c? D and △DAC are two congruent equilateral triangles.

∴∠C? DC= 180 -∠C? Da? -∠ CDA = 180-45-45 = 90, that is, c? D⊥CD

∵A? a、C? C⊥ bottom ABC, top a? b? c?

∴A? A⊥AB,C? C⊥BC,A? A⊥A? c?

∴C? D=√2,BD=√3,BC? =√5

From Pythagorean Theorem, we can know ∠C? Db = 90, which means c? D⊥BD

∴C? D⊥ plane BCD

Aircraft BC? D goes through a straight line. C? D

Aircraft BC? D⊥ plane BCD (if one plane intersects with the perpendicular of another plane, the two planes are perpendicular to each other)

(2) Solution: Know the aircraft BC? D put the prism in ABC-a? b? c? Divided into four pyramids c? ﹣A? b? BD and b-c? Computer aided design

∵C? C⊥ plane ABC

∴C? C⊥BC

∫∠ACB = 90, that is, BC⊥CA.

∴bc⊥c plane? Computer aided design

∴ Four pyramids? The volume of CAD = (1+2) ×1× (1/2 )×1× (1/3) =1/2.

Pyramid c? -A? b? BD = volume of triangular prism ABC-A? b? c? -The volume of Pyramid b-c? The volume of CAD =/kloc-0 /×/kloc-0 /× (1/2) × 2-1/2.

= 1/2

Pyramid c? -A? b? The volume of BD: Pyramid b-c? The volume of CAD =1:1.

Answer: the ratio of the two parts is pyramid C? -A? b? The volume of BD and pyramid B-C? The volume ratio of CAD is1:1.