Exploring the Area of a Triangle
2014-09-30 08:08
This GeoGebra applet explores the mathematical relationship between the base, height, and area of a triangle. Through dynamic geometric construction, users can visually observe how the area changes when altering the base length or height. Utilizing geometric objects such as polygons, vectors, segments, and angles, this model helps learners deeply understand the geometric meaning and derivation of
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.56.0" id="eacddc93-0c01-4ff9-8eac-8b826d86cc1a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="974" height="734" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2053388090349076" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="305,41,693,634" />
<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 || 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="955" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" 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="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="20"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="955" height="544"/>
<coordSystem xZero="471.1570247933881" yZero="296.9504132231406" scale="41.32231404958673" yscale="41.322314049586744"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="0" rightAngleStyle="1" checkboxSize="13" 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"/>
<grid distX="1.0" distY="1.0" distTheta="0.5235987755982988"/>
</euclidianView>
<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="30 September 2014">
<element type="point" label="A">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0" dynamicr="255 * t" dynamicg="0" dynamicb="255 * (1 - t)" colorSpace="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue[angle, Angle[u, Vector[P, A]]] SetValue[A, If[!t, P + Rotate[u, angle], A]] SetValue[B, If[!t, P + Rotate[v, angle], B]] SetValue[C, If[!t, P + Rotate[w, angle], C]]"/>
<coords x="3.5629683732953685" y="-1.9118018900982672" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0" dynamicr="255 * t" dynamicg="0" dynamicb="255 * (1 - t)" colorSpace="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue[angle, Angle[v, Vector[P, B]]] SetValue[B, If[!t, P + Rotate[v, angle], B]] SetValue[A, If[!t, P + Rotate[u, angle], A]] SetValue[C, If[!t, P + Rotate[w, angle], C]]"/>
<coords x="0.3740204323940035" y="0.8177788524738256" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0" dynamicr="255 * t" dynamicg="0" dynamicb="255 * (1 - t)" colorSpace="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue[angle, Angle[w, Vector[P, C]]] SetValue[C, If[!t, P + Rotate[w, angle], C]] SetValue[A, If[!t, P + Rotate[u, angle], A]] SetValue[B, If[!t, P + Rotate[v, angle], B]] "/>
<coords x="-4.436988805689678" y="-1.9379769623755714" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="C"/>
<output a0="poly1" a1="c" a2="a" a3="b"/>
</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="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript onUpdate="SetValue[A,A] SetValue[B,B] SetValue[C,C]"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<ggbscript val="SetValue[base_c, !base_c] SetValue[base_a, false] SetValue[base_b, false] SetLineThickness[c, If[base_c, 8, 4]] SetLineThickness[b, 4] SetLineThickness[a, 4] SetValue[o, false]"/>
<coords x="-2.729580742572093" y="-3.188947940901365" z="3.6287731573002464"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<ggbscript val="SetValue[base_a, !base_a] SetValue[base_b, false] SetValue[base_c, false] SetLineThickness[a, If[base_a, 8, 4]] SetLineThickness[b, 4] SetLineThickness[c, 4] SetValue[o, false]"/>
<coords x="2.755755814849397" y="-4.811009238083681" z="2.9036326325187867"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<ggbscript val="SetValue[base_b, !base_b] SetValue[base_a, false] SetValue[base_c, false] SetLineThickness[b, If[base_b, 8, 4]] SetLineThickness[a, 4] SetLineThickness[c, 4] SetValue[o, false]"/>
<coords x="-0.0261750722773042" y="7.9999571789850465" z="15.387594210181568"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="P">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-4" y="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-0.16666666666676866" y="-1.010666666666671" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="P" a1="A"/>
<output a0="u"/>
</command>
<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="3.729635039962137" y="-0.9011352234315961" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="P"/>
</element>
<command name="Vector">
<input a0="P" a1="B"/>
<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="0.5406870990607722" y="1.8284455191404967" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="P"/>
</element>
<command name="Vector">
<input a0="P" a1="C"/>
<output a0="w"/>
</command>
<element type="vector" label="w">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.270322139022909" y="-0.9273102957089003" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="P"/>
</element>
<element type="angle" label="angle">
<value val="0.0"/>
<show object="false" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<animation step="0.017453292519943295" speed="1" type="0" playing="false"/>
</element>
<element type="boolean" label="t">
<value val="false"/>
<show object="true" label="true"/>
<condition showObject="¬t"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="4" y="98"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[P, If[!t, (A+B+C)/3, P]] SetFixed[poly1, If[!t, true, false]] SetValue[t', !t]"/>
<caption val="改變三角形的形狀"/>
</element>
<command name="Intersect">
<input a0="OrthogonalLine[A, Line[B, C]]" a1="Line[B, C]"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="49.11948033360912" y="46.68853239717112" z="30.740000000002787"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="OrthogonalLine[B, Line[A, C]]" a1="Line[A, C]"/>
<output a0="E"/>
</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="24.51106525753591" y="-123.02121480502964" z="64.00000000000311"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="OrthogonalLine[C, Line[A, B]]" a1="Line[A, B]"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-18.34733287906787" y="35.75458478750329" z="17.619999999999475"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="D"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="false"/>
<condition showObject="base_a ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-3.430622072179168" y="-1.965067256392159" z="8.46637864896265"/>
<lineStyle thickness="8" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="高"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false"/>
<condition showObject="base_a ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.7010413296070752" y="1.223880684509206" z="-0.7386599605171134"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="F"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false"/>
<condition showObject="base_c ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.9671815473644676" y="3.3957099816789844" z="-11.021532400392152"/>
<lineStyle thickness="8" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="F"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false"/>
<condition showObject="base_c ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.2114257325150708" y="-1.415299256404697" z="1.6104997780982009"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="E"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false"/>
<condition showObject="base_b ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="39" y="70"/>
<labelMode val="3"/>
<coords x="2.7399853338023203" y="0.008964962254976505" z="-1.0321418558473177"/>
<lineStyle thickness="8" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="高"/>
</element>
<command name="Segment">
<input a0="C" a1="E"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false"/>
<condition showObject="base_b ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.01577048104707668" y="4.819974200338658" z="9.271025511634715"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="B" a1="D" a2="A"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<value val="1.5707963267948966"/>
<show object="true" label="false"/>
<condition showObject="base_a ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="2" y="18"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="20"/>
<allowReflexAngle val="false"/>
</element>
<command name="Angle">
<input a0="C" a1="F" a2="B"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<value val="1.5707963267948968"/>
<show object="true" label="false"/>
<condition showObject="base_c ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="20"/>
<allowReflexAngle val="false"/>
</element>
<command name="Angle">
<input a0="B" a1="E" a2="C"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<value val="1.5707963267948968"/>
<show object="true" label="false"/>
<condition showObject="base_b ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="20"/>
<allowReflexAngle val="false"/>
</element>
<element type="boolean" label="base_a">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="boolean" label="base_b">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="boolean" label="base_c">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<expression label="text1" exp=""底""/>
<element type="text" label="text1">
<show object="true" label="false"/>
<condition showObject="base_b"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint exp="Midpoint[b]"/>
<labelOffset x="-46" y="34"/>
</element>
<element type="boolean" label="o">
<value val="false"/>
<show object="true" label="true"/>
<condition showObject="base_a ∨ base_b ∨ base_c"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="593" y="24"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="高"/>
</element>
<expression label="text2" exp=""底""/>
<element type="text" label="text2">
<show object="true" label="false"/>
<condition showObject="base_a"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint exp="Midpoint[a]"/>
<labelOffset x="-16" y="-21"/>
</element>
<expression label="text3" exp=""高""/>
<element type="text" label="text3">
<show object="true" label="false"/>
<condition showObject="base_b ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint exp="Midpoint[h]"/>
<labelOffset x="7" y="12"/>
</element>
<expression label="text4" exp=""高""/>
<element type="text" label="text4">
<show object="true" label="false"/>
<condition showObject="base_a ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint exp="Midpoint[d]"/>
<labelOffset x="1" y="-17"/>
</element>
<expression label="text5" exp=""底""/>
<element type="text" label="text5">
<show object="true" label="false"/>
<condition showObject="base_c"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint exp="Midpoint[c]"/>
<labelOffset x="-6" y="-13"/>
</element>
<expression label="text6" exp=""高""/>
<element type="text" label="text6">
<show object="true" label="false"/>
<condition showObject="base_c ∧ o"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint exp="Midpoint[f]"/>
<labelOffset x="-24" y="-20"/>
</element>
<element type="numeric" label="s">
<value val="0.0"/>
<show object="true" label="true"/>
<condition showObject="copy ∧ showcopy"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="5" y="42"/>
<labelMode val="3"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="100.0" x="711.0" y="103.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<caption val="轉動複製三角形"/>
</element>
<element type="boolean" label="copy">
<value val="false"/>
<show object="true" label="true"/>
<condition showObject="showcopy"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="423" y="24"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="複製三角形"/>
</element>
<expression label="text7" exp=""拖拉藍點，轉動三角形。""/>
<element type="text" label="text7">
<show object="true" label="false"/>
<condition showObject="¬t"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="8" y="79"/>
</element>
<expression label="text8" exp=""拖拉紅點，改變三角形的形狀。""/>
<element type="text" label="text8">
<show object="true" label="false"/>
<condition showObject="t"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="8" y="79"/>
</element>
<expression label="text9" exp=""點擊任何一條邊，以它作底。""/>
<element type="text" label="text9">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.4" size="8" style="1"/>
<absoluteScreenLocation x="7" y="44"/>
</element>
<element type="point" label="Q">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="0" y="-29"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="6.785333333333325" y="0.7513333333333468" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="Q + B - (A + B + C) / 3" a1="180° * s" a2="Q"/>
<output a0="B_1"/>
</command>
<element type="point" label="B_1">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.326020432394097" y="2.5797788524738436" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="Q + C - (A + B + C) / 3" a1="180° * s" a2="Q"/>
<output a0="C_1"/>
</command>
<element type="point" label="C_1">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="4" y="-3"/>
<labelMode val="0"/>
<coords x="2.515011194310416" y="-0.1759769623755536" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="Q + A - (A + B + C) / 3" a1="180° * s" a2="Q"/>
<output a0="A_1"/>
</command>
<element type="point" label="A_1">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.514968373295462" y="-0.14980189009824918" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="Q" a1="A_1" a2="B_1" a3="C_1" a4="A_1"/>
<output a0="poly2" a1="q" a2="a_1" a3="b_1" a4="c_1" a5="a_2"/>
</command>
<element type="polygon" label="poly2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<condition showObject="copy ∧ showcopy"/>
<objColor r="255" g="192" b="203" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="q">
<show object="false" label="false"/>
<objColor r="255" g="192" b="203" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.901135223431596" y="3.7296350399621367" z="-8.916701996082784"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false"/>
<condition showObject="copy ∧ showcopy"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.729580742572093" y="-3.1889479409013646" z="28.223744751529953"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b_1">
<show object="true" label="false"/>
<condition showObject="copy ∧ showcopy"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.7557558148493975" y="-4.811009238083681" z="-7.777383514810949"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c_1">
<show object="true" label="false"/>
<condition showObject="copy ∧ showcopy"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.026175072277304423" y="7.999957178985046" z="1.4736387632815962"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_2">
<show object="false" label="false"/>
<objColor r="255" g="192" b="203" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.901135223431596" y="-3.7296350399621367" z="8.916701996082784"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="showcopy" exp="base_a ∨ base_b ∨ base_c" />
<element type="boolean" label="showcopy">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
</element>
<expression label="text10" exp=""轉動及拖拉複製三角形，將兩個三角形合拼成一個平行四邊形。""/>
<element type="text" label="text10">
<show object="true" label="false"/>
<condition showObject="copy ∧ showcopy"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="396" y="77"/>
</element>
<expression label="text11" exp=""如何計算三角形的面積？""/>
<element type="text" label="text11">
<show object="true" label="false"/>
<condition showObject="copy ∧ showcopy"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="397" y="115"/>
</element>
<element type="boolean" label="t'">
<value val="true"/>
<show object="true" label="true"/>
<condition showObject="¬t'"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelOffset x="4" y="98"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[t, !t']"/>
<caption val="轉動三角形"/>
</element>
</construction>
</geogebra>