P关于(ABC)的反演点为ABC的垂心H,ADH共线知(APD)过ABC外心O,∠APD=2∠ACD(以下∠均指有向角),∠ABP+∠ACP=∠BAP+∠APB+∠CAP+∠APC=(∠BAD+∠DAP)+3∠ACB+∠CAP-3∠CBA=(∠BAD+∠ACB)+∠DAP+2∠ACB+∠CAP-3∠CBA=∠ACD+∠DAP+2∠ACB+(∠CAB-∠PAB)-3∠CBA=∠ACD+∠DAP+2∠ACB+(∠CAB-∠CBA)+∠BAP+2∠ABC=∠ACD+∠DAP+3∠ACB+∠BAD+∠DAP+2∠ABC=∠ACD+2∠DAP+3∠ACB+∠BAD+(180-2∠CBA)=∠ACD+2∠DAP+3∠ACB+∠BAD+2∠BAD=∠ACD+2∠DAP+3∠ACB+3∠BAD=∠ACD+2∠DAP+3∠ACD=2∠DAP+2∠APD=2∠ADP