已知三角形PAB的面积与周长，探究P点的可能位置
2016-01-02 05:16
本范例演示一个经典几何问题：已知△PAB的面积与周长，求满足条件的P点个数。由于AB为定线段，周长固定意味着PA+PB为定值，P的轨迹为以A、B为焦点的椭圆；面积固定意味着P到AB的距离为定值，P的轨迹为平行于AB的两条直线。通过GeoGebra构造椭圆与平行线并求交点，直观展示P点的可能位置及个数，综合运用了椭圆定义、点到直线距离、几何作图等知识点，适合高中解析几何与平面几何教学使用。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.175.0" id="3fd4c583-5326-4df5-898e-61500c9922fd" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.25" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="262,197,700,550" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 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="true" location="1" size="789" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="200" window="208,302,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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" size="300" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6 || 100001" position="1" help="true" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="789" height="467"/>
<coordSystem xZero="300.0224571418358" yZero="393.46461077332503" scale="36.05943793938911" yscale="36.05943793938916"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" 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="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<collapsed val="7,8"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label ="A" exp="A:(1, 3)" type="point"/>
<element type="point" label="A">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-30" y="5"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1.0" y="3.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label ="B" exp="B:(5, 6)" type="point"/>
<element type="point" label="B">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-1" y="-8"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="5.0" y="6.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="2"/>
<coords x="-3.0" y="4.0" z="-9.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="C" exp="B + (B - A) / 2" type="point" />
<element type="point" label="C">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.0" y="7.5" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Ellipse">
<input a0="A" a1="B" a2="C"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="0.8" y0="0.6" z0="1.0" x1="-0.6" y1="0.8" z1="1.0"/>
<matrix A0="336.0" A1="364.0" A2="1599.0" A3="-48.0" A4="-792.0" A5="-1494.0"/>
<eqnStyle style="implicit"/>
</element>
<command name="LineBisector">
<input a0="a"/>
<output a0="b"/>
</command>
<element type="line" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.0" y="-3.0" z="25.5"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="A" a1="B"/>
<output a0="d"/>
</command>
<element type="line" label="d">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.0" y="4.0" z="-9.0"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<expression label="u" exp="B - A" type="vector" />
<element type="vector" label="u">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.0" y="3.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="OrthogonalVector">
<input a0="u"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.0" y="4.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="D" exp="A + (v * 4 / 5)" type="point" />
<element type="point" label="D">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.4000000000000004" y="6.2" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="E" exp="A - (v * 4 / 5)" type="point" />
<element type="point" label="E">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.4000000000000004" y="-0.20000000000000018" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="D" a1="d"/>
<output a0="e"/>
</command>
<element type="line" label="e">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.0" y="4.0" z="-29.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="E" a1="d"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.0" y="4.0" z="11.000000000000002"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="e" a2="2"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.1318833724101407" y="8.848912529307606" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="c" a1="e" a2="1"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.9318833724101411" y="6.551087470692394" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="c" a1="f" a2="2"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.931883372410139" y="2.4489125293076035" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="c" a1="f" a2="1"/>
<output a0="I"/>
</command>
<element type="point" label="I">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.8681166275898606" y="0.15108747069239525" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="F"/>
<output a0="poly1" a1="f_1" a2="a_1" a3="b_1"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="191" g="255" b="0" alpha="0.33"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="f_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-3.0" y="4.0" z="-9.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.848912529307606" y="-2.8681166275898593" z="31.453262412077184"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="5.848912529307606" y="-1.1318833724101407" z="-2.4532624120771835"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label ="arrPad" exp="arrPad:0.3"/>
<element type="numeric" label="arrPad">
<value val="0.3"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label ="arrHeight" exp="arrHeight:0.3"/>
<element type="numeric" label="arrHeight">
<value val="0.3"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label ="arrOpen" exp="arrOpen:0.3"/>
<element type="numeric" label="arrOpen">
<value val="0.3"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label ="arrHead" exp="arrHead:0.2"/>
<element type="numeric" label="arrHead">
<value val="0.2"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="LengthArrow">
<input a0="A" a1="B" a2="arrPad" a3="arrHeight" a4="arrOpen" a5="arrHead"/>
<output a0="arr1_1" a1="arr1_2" a2="arr1_3" a3="arr1_4" a4="arr1_5"/>
</command>
<element type="point" label="arr1_1">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.2699999999999996" y="4.14" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="polyline" label="arr1_2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polyline" label="arr1_3">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="arr1_4">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.23999999999999977" y="0.17999999999999994" z="-0.7799999999999994"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="arr1_5">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.2400000000000002" y="0.1800000000000006" z="-2.2800000000000047"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""" + (Distance[A, B]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="20"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="1"/>
<startPoint exp="arr1_1"/>
<labelOffset x="-5" y="17"/>
</element>
<command name="Point">
<input a0="e"/>
<output a0="J"/>
</command>
<element type="point" label="J">
<show object="false" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-3.5851419438636767" y="4.561143542102243" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<command name="OrthogonalLine">
<input a0="J" a1="e"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.0" y="-3.0" z="-0.6571371491479798"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="d" a1="g"/>
<output a0="K"/>
</command>
<element type="point" label="K">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-29.62854859659192" y="34.02858855255606" z="25.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="LengthArrow">
<input a0="J" a1="K" a2="arrPad" a3="arrHeight" a4="arrOpen" a5="arrHead"/>
<output a0="arr2_1" a1="arr2_2" a2="arr2_3" a3="arr2_4" a4="arr2_5"/>
</command>
<element type="point" label="arr2_1">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.7451419438636764" y="2.6911435421022425" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="polyline" label="arr2_2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polyline" label="arr2_3">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="arr2_4">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.1800000000000006" y="-0.2400000000000002" z="1.7400000000000055"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="arr2_5">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.17999999999999994" y="-0.24" z="0.5399999999999998"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text2" exp=""" + (Distance[e, d]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="1"/>
<startPoint exp="arr2_1"/>
<labelOffset x="-10" y="5"/>
</element>
<expression label="text3" exp=""A""/>
<element type="text" label="text3">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<startPoint exp="A"/>
<labelOffset x="-29" y="6"/>
</element>
<expression label="text4" exp=""B""/>
<element type="text" label="text4">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<startPoint exp="B"/>
<labelOffset x="4" y="-20"/>
</element>
<expression label="text5" exp=""P""/>
<element type="text" label="text5">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<startPoint exp="F"/>
<labelOffset x="-11" y="-12"/>
</element>
</construction>
</geogebra>