Math Problem 23(2) from Wuhan No.2 High School 2020 Enrollment Exam
2020-07-11 09:13
This applet demonstrates the geometric construction and solution process for Problem 23(2) from the 2020 enrollment exam of Wuhan No.2 High School. It dynamically illustrates the construction of polygons, angle bisectors, rotated segments, and intersections using GeoGebra, helping to visually understand the geometric relationships and proof strategies in the problem.
In collections
武汉市分配生考试
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.592.0" app="classic" platform="w" id="D6CCC936-C30C-4DC2-B7CB-F4492D5D435E" xsi:noNamespaceSchemaLocation="https://ggb123.cn/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1200" height="696" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="1" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="1200" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1" size="22" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="380" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="509" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" window="50,50,500,500" />
</views>
<toolbar show="true" 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="true" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1199" height="599"/>
<coordSystem xZero="95.01060941325842" yZero="531.1702081179518" scale="36.89156395594688" yscale="36.89156395594665"/>
<evSettings axes="false" 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="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<spreadsheetView>
<size width="0" height="0"/>
<prefCellSize width="70" height="0"/>
<selection hScroll="0" vScroll="0" column="-1" row="-1"/>
</spreadsheetView>
<algebraView>
<mode val="1"/>
<collapsed val="0"/>
</algebraView>
<kernel>
<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="20200711武汉二中2020年分配生考试第23题（2）" author="" date="">
<expression label="C" exp="(0, 0)" type="point"/>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-28" y="19"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="angC" exp="asind(7^(0.5) / 4)"/>
<element type="angle" label="angC">
<angleStyle val="3"/>
<value val="0.7227342478134157"/>
<show object="false" label="true"/>
<objColor r="153" g="51" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation step="0.017453292519943295" speed="1" type="0" playing="false"/>
</element>
<expression label="lBC" exp="lBC(x) = (tan(angC) * x)" />
<element type="function" label="lBC">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="Intersect">
<input a0="lBC" a1="Circle[C, 10]" a2="2"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.499999999999999" y="6.614378277661476" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="Circle[B, 8]" a1="y = 0" a2="2"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="3" y="26"/>
<labelMode val="0"/>
<coords x="12" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="C" a1="B" a2="A"/>
<output a0="angABC"/>
</command>
<element type="angle" label="angABC">
<angleStyle val="0"/>
<value val="1.445468495626831"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 1"/>
<objColor r="255" g="85" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-8" y="23"/>
<labelMode val="3"/>
<decoration type="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="2α"/>
</element>
<command name="Point">
<input a0="Segment[B, C]"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-14" y="2"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript val=""/>
<coords x="5.058759533877764" y="4.4614065563725305" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="Line[A, E]" a1="(-angABC)" a2="E"/>
<output a0="lEF"/>
</command>
<element type="line" label="lEF">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-7.444474343418127" y="3.558759533877763" z="21.78273244247935"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="lEF" a1="Segment[A, C]"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-5" y="28"/>
<labelMode val="0"/>
<coords x="261.3927893097522" y="0" z="89.33369212101752"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="E"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="true" ev="4"/>
<condition showObject="hint ≥ 1"/>
<objColor r="244" g="0" b="153" alpha="0"/>
<layer val="0"/>
<labelOffset x="-43" y="9"/>
<labelMode val="9"/>
<coords x="2.152971721288946" y="-2.4412404661222356" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="x"/>
</element>
<command name="Segment">
<input a0="C" a1="F"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 1"/>
<objColor r="244" g="0" b="153" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="40"/>
<labelMode val="9"/>
<coords x="0" y="2.926026934559602" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="y"/>
</element>
<command name="Segment">
<input a0="C" a1="B"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6.614378277661476" y="7.499999999999999" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="A"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.614378277661476" y="4.500000000000001" z="-79.37253933193772"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0" y="-12" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="A"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.4614065563725305" y="6.941240466122236" z="-53.536878676470366"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="F"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.4614065563725305" y="-2.1327325993181616" z="-13.054195749966825"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="LineBisector">
<input a0="C" a1="F"/>
<output a0="m"/>
</command>
<element type="line" label="m">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.926026934559602" y="0" z="4.28081681088413"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="m" a1="h"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 3"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-19" y="-2"/>
<labelMode val="0"/>
<coords x="32.106126081630975" y="28.314941724560068" z="21.94520200919701"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="hint">
<value val="6"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="6" absoluteScreenLocation="true" width="200" x="660" y="225" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<animation step="1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="if(hint>=3,SetValue(E,(5.06,4.46)))"/>
</element>
<expression label="text1" exp=""23. 如图, BD是ΔABC的角平分线, ∠ABC＝2∠C, 点E是边BC上的一个动点(不与B、C重合) , \\AE与BD相交于点G, F是AC边上一点, 且∠AEF＝∠ABC. \\(1) 求证：AB·CF=CE·BG; \\\textcolor{red}{(2) 若AB＝8, BC＝10, sinC＝\frac{\sqrt{7}}{4}, 设BE＝x, CF＝y, 求y与x之间的函数关系式, 并写出x的取值范围; }\\(3) 在(2) 的条件下, 当ΔAEF是以AE为腰的等腰三角形时, 求BE的长. ""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="48" y="62"/>
</element>
<expression label="text2" exp=""武汉二中2020年分配生考试""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="153" b="204" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="0"/>
<absoluteScreenLocation x="280" y="19"/>
</element>
<command name="AngularBisector">
<input a0="A" a1="B" a2="C"/>
<output a0="p"/>
</command>
<element type="line" label="p">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.9921567416492213" y="-0.12499999999999993" z="-6.614378277661475"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="p" a1="j"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<condition showObject="hint < 2"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="79.3725393319377" y="0" z="11.905880899790656"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="k" a1="p"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="true" ev="4"/>
<condition showObject="hint < 2"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="52.60409999370252" y="23.607544491429564" z="7.444474343418126"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="q"/>
</command>
<element type="segment" label="q">
<show object="true" label="false" ev="4"/>
<condition showObject="hint < 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.614378277661476" y="-0.833333333333333" z="-44.095855184409835"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A" a1="C" a2="E"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="0.7227342478134157"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 1"/>
<objColor r="153" g="51" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="2" y="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="F" a1="E" a2="A"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="1.4454684956268313"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 1"/>
<objColor r="255" g="85" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-4" y="28"/>
<labelMode val="3"/>
<decoration type="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="2α"/>
</element>
<command name="Angle">
<input a0="D" a1="B" a2="A"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="0"/>
<value val="0.7227342478134156"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≟ 1"/>
<objColor r="153" g="51" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="7" y="32"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="55"/>
<caption val="α"/>
</element>
<command name="Angle">
<input a0="C" a1="E" a2="F"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="0"/>
<value val="0.4021359692638916"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 1"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-15" y="23"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="β"/>
</element>
<command name="Angle">
<input a0="B" a1="A" a2="G"/>
<output a0="ε"/>
</command>
<element type="angle" label="ε">
<angleStyle val="0"/>
<value val="0.40213596926389167"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 1"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-22" y="-17"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="β"/>
</element>
<command name="Angle">
<input a0="F" a1="M" a2="E"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="0"/>
<value val="1.4454684956268313"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 3"/>
<objColor r="255" g="85" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="21" y="-2"/>
<labelMode val="3"/>
<decoration type="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="2α"/>
</element>
<command name="Angle">
<input a0="M" a1="F" a2="C"/>
<output a0="η"/>
</command>
<element type="angle" label="η">
<angleStyle val="0"/>
<value val="0.7227342478134156"/>
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 3"/>
<objColor r="153" g="51" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-6" y="0"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="α"/>
</element>
<expression label="text3" exp=""10""/>
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<condition showObject="hint ≥ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="1.9513781164904282" y="4.829565055325668" z="1"/>
</element>
<expression label="text4" exp=""8""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<condition showObject="hint ≥ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="10.001999129865068" y="4.666926246974665" z="1"/>
</element>
<command name="Polygon">
<input a0="E" a1="M" a2="F"/>
<output a0="t1" a1="f_1" a2="e" a3="m_1"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 3"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="f_1">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.171149956652302" y="-3.595746066597963" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.2902565997202287" y="1.463013467279801" z="-3.775325563274676"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="m_1">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-4.4614065563725305" y="2.1327325993181616" z="13.054195749966825"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="B" a1="E" a2="A"/>
<output a0="t2" a1="a" a2="b" a3="e_1"/>
</command>
<element type="polygon" label="t2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 3"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="a">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.152971721288946" y="-2.4412404661222356" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4.4614065563725305" y="6.941240466122236" z="-53.536878676470366"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e_1">
<show object="false" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-6.614378277661476" y="-4.500000000000001" z="79.37253933193772"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="m" a1="j"/>
<output a0="N"/>
</command>
<element type="point" label="N">
<show object="true" label="true" ev="4"/>
<condition showObject="hint ≥ 4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-13" y="27"/>
<labelMode val="0"/>
<coords x="51.36980173060957" y="0" z="35.112323214715225"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="M" a1="N"/>
<output a0="n"/>
</command>
<element type="segment" label="n">
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.2902565997202287" y="0" z="-1.887662781637338"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="F" a1="N" a2="M"/>
<output a0="θ"/>
</command>
<element type="angle" label="θ">
<angleStyle val="0"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<condition showObject="hint ≥ 4"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="15"/>
</element>
<expression label="text5" exp=""\frac{y}{2}""/>
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<condition showObject="hint ≥ 5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="0.3792029690973989" y="-0.022492727145939417" z="1"/>
</element>
<expression label="text6" exp=""\frac{y}{2}""/>
<element type="text" label="text6">
<show object="true" label="false" ev="8"/>
<condition showObject="hint ≥ 5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="1.870058712314922" y="-0.0089394931166914" z="1"/>
</element>
<expression label="text7" exp=""\frac{2y}{3}""/>
<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="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="0.02681888433689078" y="1.7258744626273457" z="1"/>
</element>
<expression label="text8" exp=""\frac{2y}{3}""/>
<element type="text" label="text8">
<show object="true" label="false" ev="8"/>
<condition showObject="hint ≥ 5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="1.680313435905418" y="1.5903421223348426" z="1"/>
</element>
<expression label="text9" exp=""10-x-\frac{2y}{3}""/>
<element type="text" label="text9">
<show object="true" label="false" ev="8"/>
<condition showObject="hint ≥ 5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="1.3008228830864117" y="3.325156078078882" z="1"/>
</element>
<expression label="text10" exp=""（2）解：作FC的垂直平分线交BC于点M，交FC于点N，\\则有NC＝FN＝\frac{1}{2}FC＝\frac{y}{2}．\\在RtΔMNC中，sinC＝\frac{\sqrt{7}}{4}，cosC＝\frac{3}{4}，\frac{NC}{MC}＝\frac{3}{4},则MC＝\frac{2y}{3}．\\∵MN垂直平分FC，∴MF＝MC＝\frac{2y}{3}．\\∴∠MFC＝∠C．∴∠FME＝∠MFC+∠C＝2∠C．\\∵∠ABC＝2∠C，∴∠ABC＝∠FME．\\∵∠ABE＝∠FME，∠BAE＝∠FEM，\\∴ΔABE∽ΔEMF．∴AB•MF＝BE•EM．\\∵BE＝x，BC＝10，MC＝\frac{2y}{3}，\\∴EM＝10﹣x﹣\frac{2y}{3}．\\∴8×\frac{2y}{3}＝x（10﹣x﹣\frac{2y}{3}）．\\∴y＝\frac{30x-3x^2}{2x+16}．（0＜x＜10）""/>
<element type="text" label="text10">
<show object="true" label="false" ev="40"/>
<condition showObject="hint ≥ 6"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.7" size="-5" style="0"/>
<startPoint x="13.037923552417107" y="7.811276541760732" z="1"/>
</element>
</construction>
</geogebra>