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147 lines
416 KiB
147 lines
416 KiB
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x=a + r\\text{cos}\\theta \\ \r
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y=b + r\\text{sin}\\theta \r
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\\end{matrix}\\right. `},{value:`\\begin{array}{l} \r
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\\text{\u5BF9\u4E8E\u65B9\u7A0B\u5F62\u5982\uFF1A}x^{3}-1=0 \\ \r
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x_{1}=1,x_{2}= \\omega =\\frac{-1+\\sqrt{3}i}{2} \\ \r
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x_{3}= \\omega ^{2}=\\frac{-1-\\sqrt{3}i}{2} \r
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\\end{array} `},{value:`\\begin{array}{l} \r
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a\\mathop{{x}}\\nolimits^{{2}}+bx+c=0 \\ \r
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\\Delta =\\mathop{{b}}\\nolimits^{{2}}-4ac \\ \r
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\\left\\{\\begin{matrix} \r
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\\Delta \\gt 0\\text{\u65B9\u7A0B\u6709\u4E24\u4E2A\u4E0D\u76F8\u7B49\u7684\u5B9E\u6839} \\ \r
|
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\\Delta = 0\\text{\u65B9\u7A0B\u6709\u4E24\u4E2A\u76F8\u7B49\u7684\u5B9E\u6839} \\ \r
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\\Delta \\lt 0\\text{\u65B9\u7A0B\u65E0\u5B9E\u6839} \r
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\\end{matrix}\\right. \r
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\\end{array} `},{value:`\\begin{array}{l} \r
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a\\mathop{{x}}\\nolimits^{{2}}+bx+c=0 \\ \r
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\\Delta =\\mathop{{b}}\\nolimits^{{2}}-4ac \\ \r
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\\mathop{{x}}\\nolimits_{{1,2}}=\\frac{{-b \\pm \r
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\\sqrt{{\\mathop{{b}}\\nolimits^{{2}}-4ac}}}}{{2a}} \\ \r
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\\mathop{{x}}\\nolimits_{{1}}+\\mathop{{x}}\\nolimits_{{2}}=-\\frac{{b}}{{a}} \\ \r
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\\mathop{{x}}\\nolimits_{{1}}\\mathop{{x}}\\nolimits_{{2}}=\\frac{{c}}{{a}} \r
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\\end{array} `}]}]},{name:E.formatMessage({id:"markdownEditor.geometry",defaultMessage:"\u51E0\u4F55"}),value:"\\Delta A B C ",children:[{data:[{value:"\\Delta A B C "},{value:"a \\parallel c,b \\parallel c \\Rightarrow a \\parallel b "},{value:"l \\perp \\beta ,l \\subset \\alpha \\Rightarrow \\alpha \\perp \\beta"},{value:`\\left.\\begin{matrix} \r
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a \\perp \\alpha \\ \r
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b \\perp \\alpha \r
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\\end{matrix}\\right\\}\\Rightarrow a \\parallel b`},{value:"P \\in \\alpha ,P \\in \\beta , \\alpha \\cap \\beta =l \\Rightarrow P \\in l "},{value:`\\alpha \\perp \\beta , \\alpha \\cap \\beta =l,a \\subset \\alpha ,a \\perp l \r
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\\Rightarrow a \\perp \\beta `},{value:`\\left.\\begin{matrix} \r
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a \\subset \\beta ,b \\subset \\beta ,a \\cap b=P \\ \r
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a \\parallel \\partial ,b \\parallel \\partial \r
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m \\subset \\alpha ,n \\subset \\alpha ,m \\cap n=P \\ \r
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a \\perp m,a \\perp n \r
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\\text{\u76F4\u89D2\u4E09\u89D2\u5F62\u4E2D,\u76F4\u89D2\u8FB9\u957Fa,b,\u659C\u8FB9\u8FB9\u957Fc} \\ \r
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a^{2}+b^{2}=c^{2} \r
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a \\gt b,c \\gt 0 \\Rightarrow ac \\gt bc \\ \r
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a \\gt b,c \\lt 0 \\Rightarrow ac \\lt bc \r
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a \\gt b \\gt 0,n \\in N^{\\ast},n \\gt 1 \\ \r
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\\Rightarrow a^{n}\\gt b^{n}, \\sqrt[n]{a}\\gt \\sqrt[n]{b} \r
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\\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right) `},{value:`\\begin{array}{c} \r
|
|
a,b \\in R^{+} \\ \r
|
|
\\Rightarrow \\frac{a+b}{{2}}\\ge \\sqrt{ab} \\ \r
|
|
\\left( \\text{\u5F53\u4E14\u4EC5\u5F53}a=b\\text{\u65F6\u53D6\u201C}=\\text{\u201D\u53F7}\\right) \r
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\\end{array}`},{value:`\\begin{array}{c} \r
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a,b \\in R \\ \r
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|
\\Rightarrow a^{2}+b^{2}\\gt 2ab \\ \r
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|
\\left( \\text{\u5F53\u4E14\u4EC5\u5F53}a=b\\text{\u65F6\u53D6\u201C}=\\text{\u201D\u53F7}\\right) \r
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|
\\end{array}`},{value:`\\begin{array}{c} \r
|
|
H_{n}=\\frac{n}{\\sum \\limits_{i=1}^{n}\\frac{1}{x_{i}}}= \\frac{n}{\\frac{1}{x_{1}}+ \\frac{1}{x_{2}}+ \\cdots + \\frac{1}{x_{n}}} \\ G_{n}=\\sqrt[n]{\\prod \\limits_{i=1}^{n}x_{i}}= \\sqrt[n]{x_{1}x_{2}\\cdots x_{n}} \\ A_{n}=\\frac{1}{n}\\sum \\limits_{i=1}^{n}x_{i}=\\frac{x_{1}+ x_{2}+ \\cdots + x_{n}}{n} \\ Q_{n}=\\sqrt{\\sum \\limits_{i=1}^{n}x_{i}^{2}}= \\sqrt{\\frac{x_{1}^{2}+ x_{2}^{2}+ \\cdots + x_{n}^{2}}{n}} \\ H_{n}\\leq G_{n}\\leq A_{n}\\leq Q_{n} \r
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\\text{\u5219}P \\left( A \\left| B\\right. \\right) =P \\left({B}\\right) \r
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P \\left( S \\right) =1 \r
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\\forall A \\in S \\\\ \r
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P \\left( A \\right) \\ge 0 \r
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\\end{array}`},{value:"P \\left( \\bigcup \\limits_{i=1}^{n}A_{i}\\right) = \\prod \\limits_{i=1}^{n}P \\left( A_{i}\\right) "},{value:`\\begin{array}{c} \r
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|
S= \\binom{N}{n},A_{k}=\\binom{M}{k}\\cdot \\binom{N-M}{n-k} \\\\ \r
|
|
P\\left ( A_{k}\\right ) = \\frac{\\binom{M}{k}\\cdot \\binom{N-M}{n-k}}{\\binom{N}{n}} \r
|
|
\\end{array}`},{value:`\\begin{array}{c} \r
|
|
P_{n}=n! \\\\ \r
|
|
A_{n}^{k}=\\frac{n!}{\\left( n-k \\left) !\\right. \\right.} \r
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> ${Ee}-item-active,
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> ${Ee}-submenu ${Ee}-submenu-title:hover`]:{backgroundColor:"transparent"},[`${Ee}-item, ${Ee}-submenu-title`]:{transition:[`border-color ${Le}`,`background ${Le}`].join(",")},[`${Ee}-submenu-arrow`]:{display:"none"}}}},gt=He=>{let{componentCls:Ee,menuArrowOffset:Le}=He;return{[`${Ee}-rtl`]:{direction:"rtl"},[`${Ee}-submenu-rtl`]:{transformOrigin:"100% 0"},[`${Ee}-rtl${Ee}-vertical,
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> ${Ee}-submenu > ${Ee}-submenu-title`]:{height:Le,lineHeight:`${Le}px`},[`${Ee}-item-group-list ${Ee}-submenu-title,
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${Ee}-submenu-title`]:{paddingInlineEnd:St}}};var Ae=He=>{const{componentCls:Ee,iconCls:Le,itemHeight:pe,colorTextLightSolid:be,dropdownWidth:Re,controlHeightLG:nt,motionDurationMid:Pt,motionEaseOut:St,paddingXL:Et,itemMarginInline:At,fontSizeLG:Rt,motionDurationSlow:an,paddingXS:gn,boxShadowSecondary:Ht,collapsedWidth:kt,collapsedIconSize:dn}=He,en={height:pe,lineHeight:`${pe}px`,listStylePosition:"inside",listStyleType:"disc"};return[{[Ee]:{"&-inline, &-vertical":Object.assign({[`&${Ee}-root`]:{boxShadow:"none"}},un(He))},[`${Ee}-submenu-popup`]:{[`${Ee}-vertical`]:Object.assign(Object.assign({},un(He)),{boxShadow:Ht})}},{[`${Ee}-submenu-popup ${Ee}-vertical${Ee}-sub`]:{minWidth:Re,maxHeight:`calc(100vh - ${nt*2.5}px)`,padding:"0",overflow:"hidden",borderInlineEnd:0,"&:not([class*='-active'])":{overflowX:"hidden",overflowY:"auto"}}},{[`${Ee}-inline`]:{width:"100%",[`&${Ee}-root`]:{[`${Ee}-item, ${Ee}-submenu-title`]:{display:"flex",alignItems:"center",transition:[`border-color ${an}`,`background ${an}`,`padding ${Pt} ${St}`].join(","),[`> ${Ee}-title-content`]:{flex:"auto",minWidth:0,overflow:"hidden",textOverflow:"ellipsis"},"> *":{flex:"none"}}},[`${Ee}-sub${Ee}-inline`]:{padding:0,border:0,borderRadius:0,boxShadow:"none",[`& > ${Ee}-submenu > ${Ee}-submenu-title`]:en,[`& ${Ee}-item-group-title`]:{paddingInlineStart:Et}},[`${Ee}-item`]:en}},{[`${Ee}-inline-collapsed`]:{width:kt,[`&${Ee}-root`]:{[`${Ee}-item, ${Ee}-submenu ${Ee}-submenu-title`]:{[`> ${Ee}-inline-collapsed-noicon`]:{fontSize:Rt,textAlign:"center"}}},[`> ${Ee}-item,
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> ${Ee}-item-group > ${Ee}-item-group-list > ${Ee}-item,
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|
> ${Ee}-item-group > ${Ee}-item-group-list > ${Ee}-submenu > ${Ee}-submenu-title,
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> ${Ee}-submenu > ${Ee}-submenu-title`]:{insetInlineStart:0,paddingInline:`calc(50% - ${Rt/2}px - ${At}px)`,textOverflow:"clip",[`
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${Ee}-submenu-arrow,
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${Ee}-submenu-expand-icon
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`]:{opacity:0},[`${Ee}-item-icon, ${Le}`]:{margin:0,fontSize:dn,lineHeight:`${pe}px`,"+ span":{display:"inline-block",opacity:0}}},[`${Ee}-item-icon, ${Le}`]:{display:"inline-block"},"&-tooltip":{pointerEvents:"none",[`${Ee}-item-icon, ${Le}`]:{display:"none"},"a, a:hover":{color:be}},[`${Ee}-item-group-title`]:Object.assign(Object.assign({},Ie.vS),{paddingInline:gn})}}]};const Ye=He=>{const{componentCls:Ee,motionDurationSlow:Le,motionDurationMid:pe,motionEaseInOut:be,motionEaseOut:Re,iconCls:nt,iconSize:Pt,iconMarginInlineEnd:St}=He;return{[`${Ee}-item, ${Ee}-submenu-title`]:{position:"relative",display:"block",margin:0,whiteSpace:"nowrap",cursor:"pointer",transition:[`border-color ${Le}`,`background ${Le}`,`padding ${Le} ${be}`].join(","),[`${Ee}-item-icon, ${nt}`]:{minWidth:Pt,fontSize:Pt,transition:[`font-size ${pe} ${Re}`,`margin ${Le} ${be}`,`color ${Le}`].join(","),"+ span":{marginInlineStart:St,opacity:1,transition:[`opacity ${Le} ${be}`,`margin ${Le}`,`color 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