Question

已知：等邊△ABC

（1）如圖1，P為等邊△ABC外一點，且∠BPC=120°．試猜想線段BP、PC、AP之間的數量關系，并證明你的猜想；
（2）如圖2，P為等邊△ABC內一點，且∠APD=120°．求證：PA+PD+PC＞BD．

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 

Answer 1

猜想：AP=BP+PC，
（1）證明：延長BP至E，使PE=PC，連接CE，
∵∠BPC=120°，
∴∠CPE=60°，又PE=PC，
∴△CPE為等邊三角形，
∴CP=PE=CE，∠PCE=60°，
∵△ABC為等邊三角形，
∴AC=BC，∠BCA=60°，
∴∠ACB=∠PCE，
∴∠ACB+∠BCP=∠PCE+∠BCP，
即：∠ACP=∠BCE，
∴△ACP≌△BCE，
∴AP=BE，
∵BE=BP+PE，
∴AP=BP+PC．   

（2）證明：
在AD外側作等邊△AB′D，
則點P在三角形ADB′外，
∵∠APD=120°∴由（1）得PB′=AP+PD，
在△PB′C中，有PB′+PC＞CB′，
∴PA+PD+PC＞CB′，
∵△AB′D、△ABC是等邊三角形，
∴AC=AB，AB′=AD，
∠BAC=∠DAB′=60°，
∴∠BAC+∠CAD=∠DAB′+∠CAD，
即：∠BAD=∠CAB′，
∴△AB′C≌△ADB，
∴CB′=BD，
∴PA+PD+PC＞BD．