Limit of the Intersection Point
2014-08-15 09:57
This applet demonstrates a geometric limit problem. It features a blue circle centered at (1,0) with radius 1, and a red circle centered at the origin with radius OP. As point P approaches the origin, where does the intersection point R approach? The dynamic visualization intuitively shows the limit position of R, helping understand limits in geometry.
In collections
微積分
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.4" version="4.4.40.0" id="d9632b81-ad11-432e-af18-20674e559f00" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="849" height="579" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.23557126030624265" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="740,350,700,550" />
<view id="4" toolbar="0 59 || 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 | 1011 || 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="838" 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="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" 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" />
</views>
<toolbar show="true" items="0 39 59 | 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 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="true" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="838" height="442"/>
<coordSystem xZero="164.27365686053196" yZero="218.29789204776543" scale="102.71935620902399" yscale="102.71935620902406"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="0" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="192" g="192" b="192"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="0" grid="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0" positiveAxis="true"/>
</euclidianView>
<spreadsheetView>
<size width="0" height="0"/>
<prefCellSize width="70" height="21"/>
<selection hScroll="0" vScroll="0" column="0" row="0"/>
<layout showFormulaBar="false" showGrid="true" showHScrollBar="true" showVScrollBar="true" showBrowserPanel="false" showColumnHeader="true" showRowHeader="true" allowSpecialEditor="false" allowToolTips="true" equalsRequired="false" autoComplete="false"/>
</spreadsheetView>
<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="">
<command name="Semicircle">
<input a0="(0, 0)" a1="(2, 0)"/>
<output a0="e"/>
</command>
<element type="conicpart" label="e">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<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="-1.0" A5="-0.0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
<input a0="(1, 0)" a1="1"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<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="-1.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="Segment[(0, 0.01), (0, 2)]"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-16" y="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-6.938893903907228E-18" y="1.3074253675663823" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="r" exp="y(P)" />
<element type="numeric" label="r">
<value val="1.3074253675663823"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Circle">
<input a0="(0, 0)" a1="r"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<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="-1.7093610917560897" A3="0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="d" a1="e"/>
<output a0="Q" a1="" a2="" a3=""/>
</command>
<element type="point" label="Q">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.8546805458780449" y="0.9893847867506843" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="P" a1="Q"/>
<output a0="a"/>
</command>
<element type="ray" label="a">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.31804058081569797" y="0.8546805458780449" z="-1.1174310268464391"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="a" a1="xAxis"/>
<output a0="R"/>
</command>
<element type="point" label="R">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="1.1174310268464391" y="-0.0" z="0.31804058081569797"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="P" a1="R"/>
<output a0="s0seg"/>
</command>
<element type="segment" label="s0seg">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="1.3074253675663823" y="3.5134856815457196" z="-4.593620308634134"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="O">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="nSteps">
<value val="6.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<element type="numeric" label="step">
<value val="0.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="7" y="0"/>
<labelMode val="2"/>
<slider min="0" max="nSteps" absoluteScreenLocation="true" width="100.0" x="84.0" y="393.0" fixed="false" horizontal="true"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Segment">
<input a0="O" a1="P"/>
<output a0="s1seg1"/>
</command>
<element type="segment" label="s1seg1">
<show object="true" label="false"/>
<condition showObject="1 ≤ step"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="1"/>
<selectionAllowed val="false"/>
<coords x="-1.3074253675663823" y="-6.938893903907228E-18" z="0.0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O" a1="Q"/>
<output a0="s1seg2"/>
</command>
<element type="segment" label="s1seg2">
<show object="true" label="false"/>
<condition showObject="1 ≤ step"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="1"/>
<selectionAllowed val="false"/>
<coords x="-0.9893847867506843" y="0.8546805458780449" z="0.0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="s1Q" exp=""=r(\cos\theta, \sin\theta)""/>
<element type="text" label="s1Q">
<show object="true" label="false"/>
<condition showObject="1 ≤ step"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="Q"/>
<labelOffset x="14" y="-25"/>
</element>
<expression label="s1P" exp=""=(0,r)""/>
<element type="text" label="s1P">
<show object="true" label="false"/>
<condition showObject="1 ≤ step"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="P"/>
<labelOffset x="-3" y="-26"/>
</element>
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<selectionAllowed val="false"/>
<coords x="1.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="s1SegOPLabel" exp=""r""/>
<element type="text" label="s1SegOPLabel">
<show object="true" label="false"/>
<condition showObject="1 ≤ step"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="P / 2"/>
<labelOffset x="-26" y="-10"/>
</element>
<expression label="s1SegOQLabel" exp=""r""/>
<element type="text" label="s1SegOQLabel">
<show object="true" label="false"/>
<condition showObject="1 ≤ step"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="Q / 2"/>
<labelOffset x="4" y="2"/>
</element>
<command name="Angle">
<input a0="B" a1="O" a2="Q"/>
<output a0="θ"/>
</command>
<element type="angle" label="θ">
<angleStyle val="0"/>
<value val="0.8583160866139061"/>
<show object="true" label="true"/>
<condition showObject="1 ≤ step"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="19" y="-6"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<expression label="s2lineEqu" exp=""y=\left(\frac{\sin\theta -1}{\cos\theta}\right)x+r""/>
<element type="text" label="s2lineEqu">
<show object="true" label="false"/>
<condition showObject="2 ≤ step"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="390" y="22"/>
</element>
<command name="Vector">
<input a0="Midpoint[Q, R]" a1="Midpoint[Corner[s2lineEqu, 1], Corner[s2lineEqu, 2]]"/>
<output a0="s2lineEquArrow"/>
</command>
<element type="vector" label="s2lineEquArrow">
<show object="true" label="false"/>
<condition showObject="2 ≤ step"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint x="2.1962988612476613" y="0.4987426948918819" z="1.0"/>
</element>
<expression label="s3R" exp=""x=\frac{r\cdot\cos\theta}{1-\sin\theta}""/>
<element type="text" label="s3R">
<show object="true" label="false"/>
<condition showObject="3 ≤ step"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="468" y="241"/>
</element>
<command name="Segment">
<input a0="B" a1="O"/>
<output a0="s4seg1"/>
</command>
<element type="segment" label="s4seg1">
<show object="true" label="false"/>
<condition showObject="4 ≤ step"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="-1.0" z="0.0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="Q"/>
<output a0="s4seg2"/>
</command>
<element type="segment" label="s4seg2">
<show object="true" label="false"/>
<condition showObject="4 ≤ step"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<selectionAllowed val="false"/>
<coords x="-0.9893847867506843" y="-0.14531945412195513" z="0.9893847867506843"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="O" a1="Q" a2="B"/>
<output a0="s4ang"/>
</command>
<element type="angle" label="s4ang">
<angleStyle val="0"/>
<value val="0.858316086613906"/>
<show object="true" label="true"/>
<condition showObject="4 ≤ step"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-5" y="20"/>
<labelMode val="3"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
<caption val="θ"/>
</element>
<expression label="s4label1" exp=""1""/>
<element type="text" label="s4label1">
<show object="true" label="false"/>
<condition showObject="4 ≤ step"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<startPoint exp="B / 2"/>
<labelOffset x="1" y="24"/>
</element>
<expression label="s4label2" exp=""1""/>
<element type="text" label="s4label2">
<show object="true" label="false"/>
<condition showObject="4 ≤ step"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<startPoint exp="Midpoint[B, Q]"/>
<labelOffset x="8" y="11"/>
</element>
<command name="Polygon">
<input a0="B" a1="O" a2="Q"/>
<output a0="s4tri" a1="q" a2="a_1" a3="o"/>
</command>
<element type="polygon" label="s4tri">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<condition showObject="4 ≤ step"/>
<objColor r="153" g="255" b="0" alpha="0.35"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="q">
<show object="false" label="false"/>
<objColor r="153" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.0" y="-1.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="false" label="false"/>
<objColor r="153" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-0.9893847867506843" y="0.8546805458780449" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="o">
<show object="false" label="false"/>
<objColor r="153" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.9893847867506843" y="0.14531945412195513" z="-0.9893847867506843"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="image" label="s4equ1">
<file name="8e8ad90f49468064838f22c4681dadbc/equ.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="283" y="421"/> <show object="true" label="false"/>
<condition showObject="4 ≤ step"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<command name="Angle">
<input a0="Q" a1="B" a2="O"/>
<output a0="s4ang2"/>
</command>
<element type="angle" label="s4ang2">
<angleStyle val="0"/>
<value val="1.424960480361981"/>
<show object="true" label="false"/>
<condition showObject="4 ≤ step"/>
<objColor r="255" g="0" b="0" alpha="0.2"/>
<layer val="1"/>
<labelMode val="0"/>
<decoration type="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="20"/>
<selectionAllowed val="false"/>
</element>
<element type="image" label="s5equ1">
<file name="8a4d95b5367dd5976ad87ad9d5a9c302/equ2.gif"/>
<inBackground val="false"/>
<absoluteScreenLocation x="609" y="354"/> <show object="true" label="false"/>
<condition showObject="5 ≤ step"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label="s5answer" exp=""∴\theta→\frac{\pi}{2}⇒x→4""/>
<element type="text" label="s5answer">
<show object="true" label="false"/>
<condition showObject="6 ≤ step"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="618" y="378"/>
</element>
<expression label="text1" exp=""\begin{array}{rcl} r^2 & = & 1^2+1^2-2\cos(\pi-2\theta) \\ & = & 2 + 2\cos(2\theta) \\ & = & 2 + 2(2\cos^2\theta-1) \\ & = & 4\cos^2\theta \end{array} ""/>
<element type="text" label="text1">
<show object="false" label="false"/>
<condition showObject="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="1"/>
<labelOffset x="215" y="381"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="step = step +1"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<file name="/geogebra/gui/images/go-next.png"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="1"/>
<labelOffset x="37" y="379"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="step = step -1"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<file name="/geogebra/gui/images/go-previous.png"/>
</element>
</construction>
</geogebra>