证明：延长AD交⊙O于M，由于AD，BE，CF共点O，| OD |
| AD |
| S△OBC |
| S△ABC |
| OE |
| BE |
| S△OAC |
| S△BAC |
| OF |
| CF |
| S△OAB |
| S△CAB |
则
| OD |
| AD |
| OE |
| BE |
| OF |
| CF |
而
| OD |
| AD |
| R−DM |
| 2R−DM |
| R |
| 2R−DM |
| R |
| AD |
同理有，
| OE |
| BE |
| R |
| BE |
| OF |
| CF |
| R |
| CF |
代入①得，(1−
| R |
| AD |
| R |
| BE |
| R |
| CF |
所以
| 1 |
| AD |
| 1 |
| BE |
| 1 |
| CF |
| 2 |
| R |
| 1 |
| AD |
| 1 |
| BE |
| 1 |
| CF |
| 2 |
| R |
证明：延长AD交⊙O于M，由于AD，BE，CF共点O，| OD |
| AD |
| S△OBC |
| S△ABC |
| OE |
| BE |
| S△OAC |
| S△BAC |
| OF |
| CF |
| S△OAB |
| S△CAB |
| OD |
| AD |
| OE |
| BE |
| OF |
| CF |
| OD |
| AD |
| R−DM |
| 2R−DM |
| R |
| 2R−DM |
| R |
| AD |
| OE |
| BE |
| R |
| BE |
| OF |
| CF |
| R |
| CF |
| R |
| AD |
| R |
| BE |
| R |
| CF |
| 1 |
| AD |
| 1 |
| BE |
| 1 |
| CF |
| 2 |
| R |