Menh de 169 (tinh chat giao cua duong thang va duong tron)

Trong mp hilbert cung vs tien de (E), neu 1 duog thang l chua 1 diem A nam trong dtron gama thi l se cat gama tai 2 diem

CM

Gia su co duong thag l va A thuoc l sao cho A nam trong dtron gama(O,r)

*TH1: l di qua O

Khi do theo tien de su toan dang giua cac doan thang => l co dung 2 diem B,B’ nam ve 2 phia cua O sao cho B,B’ thuoc gama

*TH2: l khong di qua O

+xay dung:

-ha OI vuong goc l tai I

-tren tia doi cua tia IO lay O’: IO’~IO

-dung duong tron delta co tam O’, bk r

Ta se cm gama cat l tai 2 diem = cach cm

_delta cat gama tai 2 diem thuoc l

+cm delta cat l

_tren tia doi cua O’O ton tai duy J: O’J~r

=> J thuoc delta va O*O’*J=> r~O’J<OJ=>OJ>r

=>J nam ngoai gama

_tẻn tia O’O, ton tai !K:O’K~r

=> K thuoc delta

Co OI~O’I, ma I nam trong gama suy raOI<r va OI~O’I suy ra O’I<r

Vay O’I<O’K

Ma I,K thuoc tia O’O=> O’*I*K=>O’K khac phia so voi J

TH1 O*K*I

=> OK<OI

Ma OI<OA=> OK<OA<r=>K thuoc gama

TH2 K*O*I

=> K,I khac phia vs O

O*I*O’=> O’,I nam cung phia voi O

Tu tren suy ra K,O’ khac phia vs O

=>O’*D*K=> OK<O’K~r=> Kthuoc gama

Vay delta chua 1 diem nam trong va 1 diem nam ngoai gama

=> delta cat gama

+cm cat tai 2 diem thuoc l

_gia su delta cat gama tai B

Xet tg OBI va O’BI co

          OI=O’I

          BI chung                  =>tgOBI~O’BI=> goc OIB~O’IB

          OB=O’B

Ma O*I*O’=> gocOIB va O’IB la 2 goc bu nhau=> OO’ vuong goc IB

Ma OO’ vuong goc voi IA

=> B thuoc l

Gia su delta tuc xuc voi gama tai B

          =>O,B,O’ thang hang

          => I,B thuoc OO’ giao l, ma OO’ khac l

          =>I trung B

          => OI~OB=r

Vay delta cat gama tai dung 2 diem phan biet ( theo menh de 1.6.8)