F4 Maths (angles in circle 2

圖片參考：http://imgcld.yimg.com/8/n/HA04628698/o/701004290200613873388690.jpg


Extend DE to meet the circumference at M ,
Extend AE to meet the circumference at N ,

Let L be the point on ME such that OL 丄 ME,
Let r be the point on NE such that OR 丄 NE ,

ㄥMEA = ㄥNED (opp.ㄥ) ,

ㄥAEF = ㄥDEF (given) ,

ㄥOEL = 180° - ㄥMEA - ㄥAEF

and ㄥOEr = 180° - ㄥNED - ㄥDEF

Hence ㄥAEF = ㄥOEL

In right-angle △OLE and △ORE ,

OE = OE (common)
ㄥAEF = ㄥOEL ,

So △OLE ≅ △ORE (RHS)

and hence Or = OL.

In right-angle △OrA and △OLD

Or = OL(proved)
OA = OD (radius)

△OrA ≅ △OLD (RHS)

Hence Ar = DL

i.e. AE + Er = DE + EL

Since Er = EL (corr. sides of △OLE and △OrE)

We have AE = DE

In △ AEF and △DEF ,

AE = DE (proved)
EF = EF (common)
ㄥAEF = ㄥDEF (given)

△AEF ≅ △DEF (SAS)

Hence AF = FD

Therefore arc AF = arc FD (equal chord , equal arc)






2010-04-29 13:28:15 補充：
Line 9 :

Hence ㄥAEF = ㄥOEL

corr to

Hence ㄥOEr = ㄥOEL


Line 12 :

ㄥAEF = ㄥOEL

corr to

ㄥOEr = ㄥOEL

Sorry!!