Exploring the General's Horse Problem in Quadratic Functions
2019-09-10 03:44
This applet demonstrates solving the classic 'General's Horse' shortest path problem in quadratic functions. By constructing a coordinate system, plotting quadratic graphs, and using geometric objects, it visually illustrates using symmetry to minimize line segment sums. It integrates algebra and geometry, ideal for teaching comprehensive quadratic problems in middle school.
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.557.0" app="classic" platform="w" id="0E9DB2D6-A835-4284-97AA-9FDB7C1BC660" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="500" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.4" orientation="1" />
</panes>
<views>
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="500" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
<view id="4" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1,1" size="400" window="100,100,700,550" />
<view id="512" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="798" height="498"/>
<coordSystem xZero="523.9088204596475" yZero="389.0409823618933" scale="20.47401822752585" yscale="20.474018227525846"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="3" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="[洋葱] 二次函数中的将军饮马-下(0548)" author="" date="">
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0"/>
<layer val="5"/>
<labelOffset x="-30" y="8"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<expression label="C" exp="(12, 0)" type="point"/>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="21" g="101" b="192" alpha="0"/>
<layer val="8"/>
<labelOffset x="-7" y="3"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="12" y="0" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<expression label="MOC" exp="(O + C) / 2" type="point" />
<element type="point" label="MOC">
<show object="false" label="false"/>
<objColor r="97" g="97" b="97" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="6" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="a">
<value val="-0.25"/>
<show object="false" label="true"/>
<condition showObject="QType ≟ 1"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<layer val="5"/>
<labelMode val="1"/>
<slider min="-2" max="2" absoluteScreenLocation="true" width="100" x="582" y="35" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="k">
<value val="1"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="8"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="hint">
<value val="5"/>
<symbolic val="true" />
<show object="true" label="false"/>
<objColor r="0" g="204" b="0" alpha="0.10000000149011612"/>
<layer val="2"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="100" x="67" y="131" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="A" exp="O + (0, 5)" type="point" />
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0"/>
<layer val="9"/>
<labelOffset x="-30" y="0"/>
<labelMode val="3"/>
<coords x="0" y="5" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<expression label="Q" exp="A + (hint * (2, 0))" type="point" />
<element type="point" label="Q">
<show object="false" label="true"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10" y="5" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="wl">
<value val="-1.5"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="wr">
<value val="13.5"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="20" absoluteScreenLocation="true" width="200" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="wu">
<value val="16.5"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="20" absoluteScreenLocation="true" width="200" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="wd">
<value val="-2.5"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Polygon">
<input a0="(wl, wd)" a1="(wr, wd)" a2="(wr, wu)" a3="(wl, wu)"/>
<output a0="wn" a1="wnd" a2="wnr" a3="wnu" a4="wnl"/>
</command>
<element type="polygon" label="wn">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="204"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="1" inverseFill="true"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="wnd">
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="15" z="37.5"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="wnr">
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-19" y="0" z="256.5"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="wnu">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="-15" z="247.5"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="wnl">
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="19" y="0" z="28.5"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text8" exp=""二次函数中的将军饮马-下""/>
<element type="text" label="text8">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="102" alpha="0"/>
<bgColor r="0" g="0" b="0" alpha="255"/>
<layer val="2"/>
<labelMode val="0"/>
<absoluteScreenLocation x="14" y="26"/>
</element>
<element type="boolean" label="blFig">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelOffset x="223" y="2"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="原题"/>
</element>
<element type="button" label="bt1">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="2"/>
<labelOffset x="296" y="4"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(QType,1)
SetColor(bt1,0,0,1)
SetColor(bt2,0,0,0)
SetColor(bt0,0,0,0)
SetBackgroundColor(bt1,1,1,0)
SetBackgroundColor(bt2,1,1,1)
SetBackgroundColor(bt0,1,1,1)
"/>
<font serif="false" sizeM="2" size="16" style="0"/>
<caption val="(1)"/>
</element>
<element type="button" label="bt2">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="2"/>
<labelOffset x="345" y="4"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(QType,2)
SetColor(bt2,0,0,1)
SetColor(bt1,0,0,0)
SetColor(bt0,0,0,0)
SetBackgroundColor(bt2,1,1,0)
SetBackgroundColor(bt1,1,1,1)
SetBackgroundColor(bt0,1,1,1)
SetValue(a,1)"/>
<font serif="false" sizeM="2" size="16" style="0"/>
<caption val="(2)"/>
</element>
<expression label="QT1" exp=""抛物线过 $O(0,0), B(10,5), C(12,0)$ 三点，$M$ 在对称轴上，
求 $MO - MB$ 的最小值与 $M$ 的坐标。""/>
<element type="text" label="QT1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="239" b="213" alpha="255"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="13" y="49"/>
</element>
<element type="numeric" label="QType">
<value val="0"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="2" absoluteScreenLocation="true" width="200" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="text10" exp=""提示""/>
<element type="text" label="text10">
<show object="true" label="false" ev="40"/>
<objColor r="51" g="255" b="0" alpha="0"/>
<bgColor r="0" g="0" b="0" alpha="255"/>
<layer val="2"/>
<labelMode val="0"/>
<absoluteScreenLocation x="15" y="134"/>
</element>
<expression label="FigL" exp="(-9.5, -1)" type="point"/>
<element type="point" label="FigL">
<show object="false" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-9.5" y="-1" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="text11" exp=""探究""/>
<element type="text" label="text11">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="255" b="255" alpha="0"/>
<bgColor r="0" g="0" b="0" alpha="255"/>
<layer val="5"/>
<labelMode val="0"/>
<absoluteScreenLocation x="412" y="24"/>
</element>
<command name="Vector">
<input a0="(wl, 0)" a1="(wr, 0)"/>
<output a0="xaxis"/>
</command>
<element type="vector" label="xaxis">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="15" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint x="-1.5" y="0" z="1"/>
</element>
<command name="Vector">
<input a0="(0, wd)" a1="(0, wu)"/>
<output a0="yaxis"/>
</command>
<element type="vector" label="yaxis">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="19" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint x="0" y="-2.5" z="1"/>
</element>
<expression label="FigR" exp="(wl, -1)" type="point" />
<element type="point" label="FigR">
<show object="false" label="false" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.5" y="-1" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="button" label="bt0">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="8"/>
<labelOffset x="223" y="4"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(blFig,If(blFig,False,True))
SetColor(bt0,If(blFig,"#0000FF","#000000"))
SetBackgroundColor(bt0,If(blFig,"#FFFF00","#FFFFFF"))"/>
<font serif="false" sizeM="2" size="16" style="0"/>
<caption val="原题"/>
</element>
<command name="Textfield">
<input a0="k"/>
<output a0="InputBox1"/>
</command>
<element type="textfield" label="InputBox1">
<show object="false" label="false"/>
<objColor r="0" g="0" b="255" alpha="1"/>
<layer val="8"/>
<labelOffset x="451" y="6"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<startPoint number="0" x="-2.0101478480484674" y="2.3957110383469975" z="1"/>
<startPoint x="-2.0101478480484674" y="2.3957110383469975" z="1"/> <linkedGeo exp="k"/>
<length val="2"/>
<caption val="k"/>
</element>
<expression label="text6" exp=""直线：$y = \;\;\; x + ( \;\;\;\;\; )$""/>
<element type="text" label="text6">
<show object="false" label="false" ev="40"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="-3.0453665598194375" y="2.383811972694458" z="1"/>
</element>
<expression label="B" exp="A + (10, 0)" type="point" />
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="21" g="101" b="192" alpha="0"/>
<layer val="8"/>
<labelOffset x="-2" y="17"/>
<labelMode val="3"/>
<coords x="10" y="5" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<expression label="qf1" exp="qf1(x) = (a * (x - x(O + C) / 2)^(2)) - (a * (x(O + C) / 2)^(2))" />
<element type="function" label="qf1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="102" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="204"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="sAB"/>
</command>
<element type="segment" label="sAB">
<show object="true" label="false"/>
<objColor r="102" g="255" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="0" y="10" z="-50"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="lx" exp="x = x(MOC)" type="line" />
<element type="line" label="lx">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="1" y="0" z="-6"/>
<lineStyle thickness="5" type="30" typeHidden="1" opacity="178"/>
<eqnStyle style="user"/>
</element>
<command name="Line">
<input a0="B" a1="C"/>
<output a0="lBC"/>
</command>
<element type="line" label="lBC">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≟ 3 ∨ hint ≟ 4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5" y="2" z="-60"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="lx" a1="lBC"/>
<output a0="M'"/>
</command>
<element type="point" label="M'">
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelOffset x="8" y="20"/>
<labelMode val="3"/>
<coords x="12" y="30" z="2"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="$M %v$"/>
</element>
<command name="Point">
<input a0="lx"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≤ 3"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="8"/>
<labelOffset x="-37" y="17"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="6" y="11.431121129277987" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Segment">
<input a0="M" a1="O"/>
<output a0="sMO"/>
</command>
<element type="segment" label="sMO">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≤ 3"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="11.431121129277987" y="-6" z="0"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="M" a1="B"/>
<output a0="sMB"/>
</command>
<element type="segment" label="sMB">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≤ 3"/>
<objColor r="0" g="153" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="6.431121129277987" y="4" z="-84.31121129277986"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="M" a1="C"/>
<output a0="sMC"/>
</command>
<element type="segment" label="sMC">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 1 ∧ hint ≤ 3"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="11.431121129277987" y="6" z="-137.17345355133585"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text2" exp=""因 $M$ 在对称轴，$\textcolor{#00F}{MO}=\textcolor{#08F}{MC}$。""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<condition showObject="hint ≥ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="13" y="160"/>
</element>
<expression label="text3" exp=""根据三角不等式，$\textcolor{#08F}{MC}-\textcolor{#090}{MB} \leq \textcolor{#80F}{BC}$""/>
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<condition showObject="hint ≥ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="14" y="198"/>
</element>
<expression label="text4" exp=""最小值发生在 $M,B,C$ 共线时""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<condition showObject="hint ≥ 3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="14" y="232"/>
</element>
<command name="Segment">
<input a0="M'" a1="B"/>
<output a0="sM'B"/>
</command>
<element type="segment" label="sM'B">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 4"/>
<objColor r="0" g="255" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="10" y="4" z="-120"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""$\textcolor{#00F}{MO} - \textcolor{#0A0}{MB} = " + (LaTeX[sMO - sMB]) + "$"" />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="462" y="10"/>
</element>
<command name="Segment">
<input a0="B" a1="C"/>
<output a0="sBC"/>
</command>
<element type="segment" label="sBC">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 3"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="5" y="2" z="-60"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text7" exp=""此时 $\textcolor{#00F}{MO}-\textcolor{#090}{MB} =\textcolor{#80F}{BC}=\textcolor{#80F}{" + (LaTeX[SurdText[Distance[B, C]]]) + " }$"" />
<element type="text" label="text7">
<show object="true" label="false" ev="8"/>
<condition showObject="hint ≥ 5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="15" y="306"/>
</element>
<command name="Segment">
<input a0="M'" a1="O"/>
<output a0="sM'O"/>
</command>
<element type="segment" label="sM'O">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="15" y="-6" z="0"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text5" exp=""可解得对称轴与直线$\textcolor{red}{BC}$ 的交点 $
\textcolor{red}{M=" + (LaTeX[M']) + "}$。"" />
<element type="text" label="text5">
<show object="true" label="false" ev="8"/>
<condition showObject="hint ≥ 4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="13" y="266"/>
</element>
<element type="image" label="pic1">
<file name="6806537567ccdaff98c3525a64dca820/[洋葱] 二次函数中的将军饮马-下(0723).jpg"/>
<inBackground val="false"/>
<absoluteScreenLocation x="3" y="493"/> <show object="true" label="false"/>
<condition showObject="blFig ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="9"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
</construction>
</geogebra>