Geometric Demonstration of the AM-GM Inequality
2019-08-01 14:07
This GeoGebra applet provides a visual and geometric demonstration of the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Using geometric objects such as segments, midpoints, circles, and orthogonal lines, it constructs a model that helps users understand the relationship and geometric interpretation between the arithmetic and geometric means. Ideal for teaching and learning inequalities at the
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.553.0" app="classic" platform="d" id="db6c5827-cbe9-4eab-ae51-085cfe35c582" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1366" height="728" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.13616398243045388" 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="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="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="486" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1148" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="186" window="100,100,600,400" />
<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="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" 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="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1148" height="534"/>
<coordSystem xZero="618.1920619823718" yZero="279.15342683671986" scale="6.911670520323495" yscale="6.911670520323478"/>
<evSettings axes="false" grid="true" 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="false"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="算幾終極版" author="" date="">
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="t" exp="40"/>
<element type="numeric" label="t">
<value val="40.0"/>
<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="50" absoluteScreenLocation="true" width="200.0" x="1181.0" y="233.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<expression label="B" exp="(t, 0)" type="point" />
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="40.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Textfield">
<input a0="t"/>
<output a0="inputBox1"/>
</command>
<element type="textfield" label="inputBox1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="204" alpha="1.0"/>
<layer val="0"/>
<labelOffset x="50" y="352"/>
<labelMode val="3"/>
<auxiliary val="true"/>
<startPoint number="0" x="-82.20763132612086" y="-10.539647824513313" z="1"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint x="-82.20763132612086" y="-10.539647824513313" z="1"/> <linkedGeo exp="t"/>
<length val="3"/>
<caption val="a+b"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.0" y="40.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="f"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="20.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="C" a1="t / 2"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="96" g="96" b="96" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="0.0" A3="0.0" A4="-20.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<element type="numeric" label="a">
<value val="29.5"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="204" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="t" absoluteScreenLocation="true" width="200.0" x="227.0" y="373.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="8.0"/>
<lineStyle thickness="16" type="0" typeHidden="1"/>
<animation step="0.5" speed="1" type="0" playing="false"/>
</element>
<expression label="D" exp="(a, 0)" type="point" />
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="29.5" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="D" a1="f"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-40.0" y="-0.0" z="1180.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="g"/>
<output a0="E" a1="F"/>
</command>
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="29.5" y="-17.599715906798043" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="29.5" y="17.599715906798043" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F" a1="A"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="128" g="128" b="128" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="17.599715906798043" y="-29.5" z="-0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="B"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="128" g="128" b="128" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="17.599715906798043" y="10.5" z="-703.9886362719217"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="C" a1="f"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-40.0" y="-0.0" z="800.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="j"/>
<output a0="G" a1="H"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="20.0" y="-20.0" z="1.0"/>
<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="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="20.0" y="20.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F" a1="D"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="19" y="21"/>
<labelMode val="3"/>
<coords x="17.599715906798043" y="0.0" z="-519.1916192505422"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\sqrt{ab}$"/>
</element>
<command name="Segment">
<input a0="H" a1="C"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-43" y="-30"/>
<labelMode val="3"/>
<coords x="20.0" y="0.0" z="-400.0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\frac{a+b}{2}$"/>
</element>
<command name="Segment">
<input a0="D" a1="A"/>
<output a0="m"/>
</command>
<element type="segment" label="m">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="5" y="56"/>
<labelMode val="3"/>
<coords x="0.0" y="-29.5" z="-0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="a"/>
</element>
<command name="Segment">
<input a0="D" a1="B"/>
<output a0="n"/>
</command>
<element type="segment" label="n">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="51" b="153" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-9" y="23"/>
<labelMode val="3"/>
<coords x="0.0" y="10.5" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="b"/>
</element>
<expression label="text1" exp=""算幾不等式

若a、 \; b為非負實數，則\frac{a+b}{2}≥\sqrt{ab}""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="-85.02779932927618" y="34.73871806522845" z="1.0"/>
</element>
<expression label="text2" exp=""a+b為直徑，\frac{a+b}{2}即為半徑。""/>
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<absoluteScreenLocation x="36" y="120"/>
</element>
<expression label="text3" exp=""\sqrt{ab}為直角三角形中，斜邊上的高。""/>
<element type="text" label="text3">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<absoluteScreenLocation x="36" y="207"/>
</element>
<expression label="text4" exp=""a=" + (LaTeX[a]) + " \; \; \; b= " + (LaTeX[n]) + """ />
<element type="text" label="text4">
<show object="true" label="true" ev="40"/>
<objColor r="255" g="51" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="-82.86106830400334" y="-22.570568038301634" z="1.0"/>
</element>
<expression label="text5" exp=""\frac{a+b}{2}=" + (LaTeX[l]) + " \; \; \; \sqrt{ab}=" + (LaTeX[k]) + " "" />
<element type="text" label="text5">
<show object="true" label="true" ev="40"/>
<objColor r="255" g="51" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<absoluteScreenLocation x="331" y="419"/>
</element>
</construction>
</geogebra>