探索下列∠A与∠P之间的关系,并说明理由.
(1)如图①,BP、CP分别平分∠ABC、∠ACB;
(2)如图②,BP、CP分别平分∠ABC、∠ACB的补角:
(3)如图③,BP平分∠ABC的补角、CP平分∠ACB的补角.
(1)∵BP、CP分别平分∠ABC和∠ACB,
∴∠PBC=
∠ABC,∠PCB=
∠ACB,
∴∠PBC+∠PCB=
(∠ABC+∠ACB)=
×(180°-∠A),
∴∠P=180°-(∠PCB+∠PBC)=90°+
∠A.
(2)∠ACE=∠A+∠ABC,
∵CP平分∠ACE,BP平分∠ABC,
∴∠PBC=
∠ABC,∠PCA=
∠ACE=
∠A+
∠ABC,
∴∠P=180°-(∠PBC+∠PCA+∠ACB)=
∠A;
(3)∵∠DBC=∠A+∠ACB,∠ECB=∠A+∠ABC,
∴∠DBC+∠ECB=∠A+∠ACB+∠A+∠ABC,
∵BP、CP分别平分∠DBC和∠ECD,
∴∠PBC=
∠DBC,∠PCB=
∠ECB,
∴∠PBC+∠PCB=
(∠DBC+∠ECB),
∴∠P=180°-(∠PBC+∠PCB)=180°-
(∠DBC+∠ECB)=90°-
∠A.
故答案为:(1)∠P=90°+
∠A;(2)∠P=∠A;(3)∠P=90°-
∠A,