例一：

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解：

设:∣∣bc∣∣=BC；∣∣ad∣∣=AD；∣∣ae∣∣=AE；∣∣fc∣∣=FC；∣∣ba∣∣=ABbc=(x,0)；ad=(x,0)；ae=(0,x2)；fc=(x2,0)；ba=(0,x)得:ba+bc=bd=(x,x)ea+ad=ed=(x,−x2)db+dc=dc=(0,−x)dc+cf=df=(−x2,−x)∣∣df∣∣=[df,df]；∣∣ed∣∣=[ed,ed]∣∣ed∣∣=∣∣df∣∣=ED=DFcosθ=[df,ed]∣∣ed∣∣∣∣df∣∣cosθ=0=90o⇔∠EDF显然:DG⊥EF设:\\ ||bc||=BC；||ad||=AD；||ae||=AE；||fc||=FC；||ba||=AB\\ bc = (x,0)；ad=(x,0)；ae=(0,x_2)；fc=(x_2,0)；ba=(0,x)\\ 得:\\ ba+bc=bd=(x,x)\\ ea+ad=ed=(x,-x_2)\\ db+dc=dc=(0,-x)\\ dc+cf=df=(-x_2,-x)\\ ||df||=\sqrt{[df,df]}；||ed||=\sqrt{[ed,ed]}\\ ||ed||=||df||=ED=DF\\ cos\theta=\frac{[df,ed]}{||ed||||df||}\\ cos\theta=0=90^o\Leftrightarrow\angle EDF\\ 显然:DG\perp EF

例二：

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解：

设:∣∣bf∣∣=BF；∣∣be∣∣=BE；∣∣eo∣∣=EO；∣∣cf∣∣=CF；∣∣ba∣∣=BA；∣∣ef∣∣=EF显然且设:bf=(5,0)；be=(x,0)；cf=(x,0)；ba=(x,y)；ef=(5−x,0)得:ba+ed=ea=(0,y)ea+ef=ed=(5−x,y)eo=(5−x2,y2){∣∣ba∣∣=3∣∣eo∣∣=2解得{x=1.8y=2.4显然:y=AE=DE=2.4设:\\ ||bf||=BF；||be||=BE；||eo||=EO；||cf||=CF；||ba||=BA；||ef||=EF\\ 显然且设:\\ bf=(5,0)；be=(x,0)；cf=(x,0)；ba=(x,y)；ef=(5-x,0)\\ 得:\\ ba+ed=ea=(0,y)\\ ea+ef=ed=(5-x,y)\\ eo=(\frac{5-x}{2},\frac{y}{2})\\ \left\{ \begin{aligned} ||ba||=3 \\ ||eo||=2 \\ \end{aligned} \right. \\ 解得 \left\{ \begin{aligned} x&=1.8 \\ y&=2.4 \\ \end{aligned} \right. \\ 显然:\\ y=AE=DE=2.4

例三：

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解：

设:∣∣ba∣∣=BA；∣∣bc∣∣=BC；∣∣bd∣∣=BD得:AD//BC⇔∠DAC=∠ACB=45o⇔BA=ACba=(2,y)；bc=(22,0)[ba,ba]=2y=2ba+bc=bd∣∣bd∣∣=25设:\\ ||ba||=BA；||bc||=BC；||bd||=BD\\ 得:\\ AD//BC\Leftrightarrow\angle DAC=\angle ACB=45^o\Leftrightarrow BA=AC\\ ba=(\sqrt{2},y)；bc=(2\sqrt{2},0)\\ \begin{equation} \begin{aligned} \sqrt{[ba,ba]}&=2\\ y&=\sqrt{2} \end{aligned} \end{equation}\\ ba+bc=bd\\ ||bd||=2\sqrt{5}

例四：

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解：

显然:PC=(x−2)2+(x33)2PB=x2+(x33)22f(x)=(x−2)2+(x33)2+x2+(x33)22dfdx=(2 x−3) 4 x2−12 x+12 ∣x∣+2 x3−6 x2+6 x(2 3 x2−2⋅3 3 x+2⋅3 3) ∣x∣令:dfdx=0解得:x=0得:f(0)=2故:min(PC+PB2)=2显然:\\ PC=\sqrt{(x-2)^2+(x\frac{3}{3})^2}\\ PB=\frac{\sqrt{x^2+(x\frac{\sqrt{3}}{3})^2}}{2}\\ f(x)=\sqrt{(x-2)^2+(x\frac{3}{3})^2}+\frac{\sqrt{x^2+(x\frac{\sqrt{3}}{3})^2}}{2}\\ \frac{df}{dx}=\dfrac{\left(2\,x-3\right)\,\sqrt{4\,{x}^{2}-12\,x+12}\,\left|x\right|+2\,{x}^{3}-6\,{x}^{2}+6\,x}{\left(2\,\sqrt{3}\,{x}^{2}-2\cdot 3\,\sqrt{3}\,x+2\cdot 3\,\sqrt{3}\right)\,\left|x\right|}\\ 令:\\ \frac{df}{dx}=0\\ 解得:x=0\\ 得:\\ f(0)=2\\ 故:\\ min(PC+\frac{PB}{2})=2

虽然，例四其实连算的不用算……