摘要：042]解:(1)∵点是的中点.∴.∴． 又∵是的角平分线.∴. ∴.∴．········································································· 3分 (2)过点作的平分线的垂线.垂足为.点即为所求． 易知点的坐标为(2.2).故.作. ∵是等腰直角三角形.∴. ∴点的坐标为(3.3)． ∵抛物线经过原点.∴设抛物线的解析式为． 又∵抛物线经过点和点.∴有 解得 ∴抛物线的解析式为．··············································································· 7分 (3)由等腰直角三角形的对称性知D点关于的平分线的对称点即为点． 连接.它与的平分线的交点即为所求的点(因为.而两点之间线段最短).此时的周长最小． ∵抛物线的顶点的坐标.点的坐标. 设所在直线的解析式为.则有.解得． ∴所在直线的解析式为． 点满足.解得.故点的坐标为． 的周长即是． (4)存在点.使．其坐标是或．································· 14分

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