百度作业网一道圆的题
来源：学生作业帮 编辑：百度作业网作业帮 分类：数学作业 时间：2024/07/09 05:36:46
一道圆的题
证明:(1)连接I1A2和I1A3,
∵I1是△A1A2A3的内心,
∴∠A2I1A3=90°+1/2∠A2A1A3,
同理,I2是△A4A2A3的内心,
∴∠A2I2A3=90°+1/2∠A2A4A3,
又∵同弧所对的圆周角相等,
∴∠A2A1A3=∠A2A4A3
∴∠A2I1A3=∠A2I2A3
故A2、I1、I2、A3四点共圆.
(2)连接I1A2和I2A3,I3A4,
由(1)知A2、I1、I2、A3四点共圆.
∴∠I1A2A3+∠I1I2A3=180°
同理可证,A3、I2、I3、A4四点共圆.
∴∠I3A4A3+∠I3I2A3=180°
又∵∠I1I2A3+∠I3I2A3=360°-∠I1I2I3,
∴∠I1I2I3=∠I1A2A3+∠I3A4A3
=1/2∠A1A2A3+1/2∠A1A4A3
=1/2(∠A1A2A3+∠A1A4A3)
=1/2*180°
=90°.
∵I1是△A1A2A3的内心,
∴∠A2I1A3=90°+1/2∠A2A1A3,
同理,I2是△A4A2A3的内心,
∴∠A2I2A3=90°+1/2∠A2A4A3,
又∵同弧所对的圆周角相等,
∴∠A2A1A3=∠A2A4A3
∴∠A2I1A3=∠A2I2A3
故A2、I1、I2、A3四点共圆.
(2)连接I1A2和I2A3,I3A4,
由(1)知A2、I1、I2、A3四点共圆.
∴∠I1A2A3+∠I1I2A3=180°
同理可证,A3、I2、I3、A4四点共圆.
∴∠I3A4A3+∠I3I2A3=180°
又∵∠I1I2A3+∠I3I2A3=360°-∠I1I2I3,
∴∠I1I2I3=∠I1A2A3+∠I3A4A3
=1/2∠A1A2A3+1/2∠A1A4A3
=1/2(∠A1A2A3+∠A1A4A3)
=1/2*180°
=90°.