几何综合题：圆、多边形旋转与轨迹探究
2019-04-22 15:36
本范例是一道综合性平面几何作图题，利用 GeoGebra 演示了线段、圆、圆弧、扇形、多边形等几何对象的构造过程。通过旋转（Rotate）与缩放（Dilation）等变换操作，结合向量与角度的度量，动态追踪关键点在变换下的运动轨迹（Locus）。该范例涵盖圆与直线的交点求解、圆弧与扇形的绘制、多边形顶点变换及轨迹生成，适用于探究平面几何中旋转变换与轨迹问题，帮助学生直观理解几何变换的性质与规律。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.534.0" app="classic" platform="w" id="7efdee42-1948-451b-97a6-03e0f03e8b92" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="525" height="437" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.4588910133843212" 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 39 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="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="523" height="435"/>
<coordSystem xZero="241.99999999999997" yZero="351.99999999999983" scale="49.99999999999998" yscale="49.999999999999986"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="1" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<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"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0" y="2" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A + (3, 3)" a1="A + (3, -3)"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6" y="0" z="-18"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A - (3, 3)" a1="A + (3, 3)"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6" y="6" z="-12"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A + (-3, 3)" a1="A + (3, -3)"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6" y="6" z="-12"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="3" y="2" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="A" a1="B"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-5" A3="0" A4="0" A5="-2"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="c" a1="g"/>
<output a0="G" a1="H"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.1213203435596424" y="-0.12132034355964239" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="H">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.121320343559643" y="4.121320343559643" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="H" a1="A + (3, 3)"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8786796564403572" y="0.8786796564403572" z="-1.7573593128807161"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="c" a1="f"/>
<output a0="I" a1="J"/>
</command>
<element type="point" label="I">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.1213203435596424" y="4.121320343559642" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="J">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.121320343559643" y="-0.12132034355964283" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="J" a1="A + (3, -3)"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.8786796564403572" y="0.8786796564403572" z="-1.7573593128807143"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CircleArc">
<input a0="A" a1="J" a2="H"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-5.0000000000000036" A3="0" A4="0" A5="-2"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-3" y="2.0000000000000004" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1.8369701987210297e-16" y="-1" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="t">
<value val="0"/>
<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="1" absoluteScreenLocation="true" width="200" x="516" y="221" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="5" type="3" playing="false"/>
</element>
<expression label="text1" exp=""計算下圖著色部分的面積。""/>
<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"/>
<absoluteScreenLocation x="21" y="38"/>
</element>
<element type="boolean" label="a">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="348" y="10"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[t, 0]
StartAnimation[t, a]"/>
<caption val="提示"/>
</element>
<command name="Polygon">
<input a0="{A - (3, 3), A + (3, -3), A + (3, 3), A + (-3, 3)}"/>
<output a0="q1"/>
</command>
<element type="polygon" label="q1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Point">
<input a0="{d, i, l, h}"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.1213203435596424" y="-0.12132034355964327" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="E'" exp="E" type="point" />
<element type="point" label="E'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.1213203435596424" y="-0.12132034355964327" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Locus">
<input a0="E'" a1="E"/>
<output a0="loc1"/>
</command>
<element type="locus" label="loc1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.30000001192092896"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="F" exp="A + 3 / sqrt(2)" type="point" />
<element type="point" label="F">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.1213203435596424" y="4.121320343559642" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="F" a1="((-t) * 90°)" a2="A"/>
<output a0="F'"/>
</command>
<element type="point" label="F'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.1213203435596424" y="4.121320343559642" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="CircleSector">
<input a0="A" a1="F'" a2="Rotate[F', 90°, A]"/>
<output a0="e"/>
</command>
<element type="conicpart" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.30000001192092896"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-4.999999999999998" A3="0" A4="0" A5="-2"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A + (3 / sqrt(2), 3 / sqrt(2))" a1="A" a2="A + (-3 / sqrt(2), 3 / sqrt(2))"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="1.570796326794897"/>
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Segment">
<input a0="A + (3.2, -3)" a1="A + (3.6, -3)"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0.3999999999999999" z="0.3999999999999999"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A + (3.2, 3)" a1="A + (3.6, 3)"/>
<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="0" y="0.3999999999999999" z="-2"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Vector">
<input a0="Midpoint[j]" a1="Midpoint[k]"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="6" z="0"/>
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint x="3.4000000000000004" y="-1" z="1"/>
</element>
<command name="Vector">
<input a0="Midpoint[k]" a1="Midpoint[j]"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="0" y="9"/>
<labelMode val="3"/>
<coords x="0" y="-6" z="0"/>
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint x="3.4000000000000004" y="5" z="1"/>
<caption val="10 cm"/>
</element>
<command name="Rotate">
<input a0="e" a1="180°" a2="A"/>
<output a0="e'"/>
</command>
<element type="conicpart" label="e'">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.30000001192092896"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="-1" y0="1.2246467991473532e-16" z0="1.0" x1="-1.2246467991473532e-16" y1="-1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-4.999999999999998" A3="0" A4="0" A5="-2"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CircleArc">
<input a0="A" a1="I" a2="G"/>
<output a0="p"/>
</command>
<element type="conicpart" label="p">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-4.999999999999998" A3="0" A4="0" A5="-2"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="G" a1="A - (3, 3)"/>
<output a0="m"/>
</command>
<element type="segment" label="m">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.8786796564403576" y="-0.8786796564403576" z="1.7573593128807152"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="I" a1="A + (-3, 3)"/>
<output a0="n"/>
</command>
<element type="segment" label="n">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8786796564403581" y="-0.8786796564403576" z="1.7573593128807143"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A + (-3, 3)" a1="A - (3, 3)"/>
<output a0="q"/>
</command>
<element type="segment" label="q">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6" y="0" z="18"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="{p, n, q, m}"/>
<output a0="K"/>
</command>
<element type="point" label="K">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2.1213203435596424" y="-0.12132034355964239" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="K'" exp="K" type="point" />
<element type="point" label="K'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.1213203435596424" y="-0.12132034355964239" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Locus">
<input a0="K'" a1="K"/>
<output a0="loc2"/>
</command>
<element type="locus" label="loc2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.30000001192092896"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
</construction>
</geogebra>