正方形的内切圆

hbghlyj

正方形ABCD内切圆于M的切线交BC,CD于P,Q(1)求证:AF∥ME
作Q关于AP中垂线的对称点Q'，(2)求证:APQQ'共圆，且圆心的轨迹是以C为圆心的圆弧，两端为OB,OD中点
(3)求证:AQ,FQ'交点的轨迹是线段
AO'交PQ于I,(4)求证：I的轨迹是双曲线的一支

hbghlyj

(1)设正方形边长为2,$\therefore {\text{PM}} = {\text{PF}},{\text{QG}} = {\text{QM}}$,在${\text{Rt}}\vartriangle {\text{PCQ}}$中,${\text{P}}{{\text{Q}}^2} = {\text{P}}{{\text{C}}^2} + {\text{C}}{{\text{Q}}^2},\therefore {\left( {{\text{PF}} + {\text{QG}}} \right)^2} = {\left( {1 - {\text{PF}}} \right)^2} + {\left( {1 - QG} \right)^2},\therefore PF\cdot{\text{QG}} = 1 - {\text{PF}} - {\text{QG}},{\text{BP}}\cdot{\text{DQ}} = \left( {1 + {\text{PF}}} \right)\left( {1 + {\text{QG}}} \right) = 2 = {\text{BE}}\cdot{\text{AD}},\vartriangle {\text{ADQ}} \sim \vartriangle {\text{PBE}},{\text{AP}}\parallel {\text{HQ}}$

isee

回复 1# hbghlyj


这么大的点，看着超级不习惯，闪了闪了

hbghlyj

回复 3# isee
同问。。我粘贴到word里只有几寸，为什么到这里变这么大？