Converse of the Angle in a Semicircle Theorem
2024-12-27 02:20
This GeoGebra applet demonstrates the converse of the angle in a semicircle theorem (also known as the converse of Thales's theorem). It shows that if an inscribed angle is a right angle, then the chord it subtends must be a diameter of the circle. Through dynamic geometric construction, users can drag points to observe the relationships between angles and segments, verifying the property that a r
In collections
Basic Properties of Circles
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.2.871.0" app="classic" platform="w" id="4863d922-ff0c-425b-aef0-a1b40534e39d" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/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="0.9975" orientation="1" />
</panes>
<views>
<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="3" size="798" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="200" tab="ALGEBRA" window="487,5,307,741" />
<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="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="390" window="165,101,952,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" 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="false" 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="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="797" height="597"/>
<coordSystem xZero="377.2262825235985" yZero="251.0099706655488" scale="50.53519054974261" yscale="50.58897443111344"/>
<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>
<spreadsheetView>
<dimensions rows="100" columns="26"/></spreadsheetView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="0" max="0" step="0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="Q">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-22" y="29"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<coords x="-2.9331299815263767" y="-1.6997780702501326" z="1"/>
</element>
<element type="point" label="P">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-20" y="-5"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<coords x="-1.4094392788227303" y="2.7675985180565164" z="1"/>
</element>
<element type="point" label="R">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="3" y="22"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<coords x="1.2776401778868829" y="-1.675069508009443" z="1"/>
</element>
<element type="point" label="D_1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<coords x="0.15382780576932095" y="2.708297058919707" z="1"/>
</element>
<command name="Circle">
<input a0="P" a1="Q" a2="R"/>
<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="4" type="0" typeHidden="1"/>
<eqnStyle style="specific"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-6.845176406160597" A3="0" A4="0.8381113745512885" A5="-0.07920405473819264"/>
</element>
<command name="Center">
<input a0="c"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<pointSize val="5"/>
<pointStyle val="1"/>
<coords x="-0.8381113745512885" y="0.07920405473819264" z="1"/>
</element>
<command name="Radius">
<input a0="c"/>
<output a0="r"/>
</command>
<element type="numeric" label="r">
<value val="2.74843234673874"/>
</element>
<command name="Sequence">
<input a0="Rotate[O - (0, r), (2 * i°), O]" a1="i" a2="91" a3="179"/>
<output a0="l1"/>
</command>
<element type="list" label="l1">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Point">
<input a0="l1"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-23" y="30"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="-3.264833099239871" y="-1.2111067742965358" z="1"/>
</element>
<command name="Intersect">
<input a0="c" a1="Line[A, D_1]" a2="2"/>
<output a0="D'"/>
</command>
<element type="point" label="D'">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<coords x="0.1105787698626477" y="2.6587131891456828" z="1"/>
</element>
<command name="If">
<input a0="Distance[D_1, D'] < 0.15" a1="true" a2="false"/>
<output a0="near"/>
</command>
<element type="boolean" label="near">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<caption val="P"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="-0.47345832179351416" y="2.80333850731469" z="1"/>
</element>
<command name="Sequence">
<input a0="Rotate[O - (0, r), (2 * i°), O]" a1="i" a2="1" a3="89"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Point">
<input a0="l2"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-1" y="33"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="
"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="1.6727065114498811" y="-1.0386840916952662" z="1"/>
</element>
<command name="Angle">
<input a0="A" a1="O" a2="B"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="2.2340214425527423"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="35"/>
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-32" y="29"/>
<labelMode val="9"/>
<caption val="q"/>
</element>
<command name="Angle">
<input a0="A" a1="D" a2="B"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<value val="1.1170107212763711"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="35"/>
<show object="true" label="true" ev="4"/>
<condition showObject="β ≠ 90°"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-18" y="37"/>
<labelMode val="2"/>
<caption val="p"/>
</element>
<expression label="text1" exp=""拖拉 A 或 B 點,令 ∠APB = 90°。你有甚麼發現?""/>
<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="11" y="27"/>
</element>
<expression label="text2" exp=""\boxed\text{若 ∠APB = 90°,則 AB 是直徑。\\ \\ \\
【 簡寫:半圓上的圓周角的逆定理 】}""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<condition showObject="f ∧ β ≟ 90°"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="38" y="441"/>
</element>
<element type="boolean" label="f">
<value val="false"/>
<show object="true" label="true"/>
<condition showObject="β ≟ 90°"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="39" y="409"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="定理:"/>
</element>
<command name="If">
<input a0="D ≟ D'" a1="true" a2="false"/>
<output a0="located"/>
</command>
<element type="boolean" label="located">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="If">
<input a0="α ≟ β" a1="true" a2="false"/>
<output a0="equalangle"/>
</command>
<element type="boolean" label="equalangle">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="Segment">
<input a0="A" a1="O"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="-1.2903108290347285" y="2.4267217246885826" z="-1.2736303828373046"/>
</element>
<command name="Segment">
<input a0="O" a1="B"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="1.117888146433459" y="2.5108178860011696" z="0.7380478137214685"/>
</element>
<command name="Segment">
<input a0="A" a1="D"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="-4.014445281611225" y="2.791374777446357" z="-9.725840927925885"/>
</element>
<command name="Segment">
<input a0="D" a1="B"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="3.8420225990099564" y="2.1461648332433954" z="-4.197388948055811"/>
</element>
<command name="Intersect">
<input a0="c" a1="Ray[O, Midpoint[Segment[A, B]]]"/>
<output a0="M" a1=""/>
</command>
<element type="point" label="M">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-1" y="26"/>
<labelMode val="0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<coords x="-0.7421924689292331" y="-2.667554021275071" z="1"/>
</element>
<command name="Mirror">
<input a0="M" a1="O"/>
<output a0="N"/>
</command>
<element type="point" label="N">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<coords x="-0.9340302801733439" y="2.8259621307514564" z="1"/>
</element>
<command name="If">
<input a0="α ≠ 180°" a1="CircumcircleArc[A, N, B]" a2="Semicircle[A, B]"/>
<output a0="c1"/>
</command>
<element type="conicpart" label="c1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<eqnStyle style="specific"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-6.845176406160598" A3="0" A4="0.8381113745512883" A5="-0.07920405473819307"/>
</element>
<command name="If">
<input a0="α ≠ 180°" a1="CircumcircleArc[A, M, B]" a2="Semicircle[B, A]"/>
<output a0="c2"/>
</command>
<element type="conicpart" label="c2">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-10" y="32"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<eqnStyle style="specific"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-6.845176406160594" A3="0" A4="0.8381113745512885" A5="-0.07920405473819213"/>
</element>
<command name="Distance">
<input a0="D'" a1="c1"/>
<output a0="d1"/>
</command>
<element type="numeric" label="d1">
<value val="1.4043333874306805e-15"/>
</element>
<command name="Distance">
<input a0="D'" a1="c2"/>
<output a0="d2"/>
</command>
<element type="numeric" label="d2">
<value val="4.013849739764367"/>
</element>
<expression label="atopparc" exp="located ∧ (d2 ≟ 0 ∧ α ≤ 180° ∨ α > 180° ∧ d1 ≟ 0)" />
<element type="boolean" label="atopparc">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="Angle">
<input a0="B" a1="O" a2="A"/>
<output a0="α'"/>
</command>
<element type="angle" label="α'">
<angleStyle val="0"/>
<value val="4.049163864626844"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="35"/>
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="9"/>
<caption val="q"/>
</element>
<command name="Angle">
<input a0="A" a1="D" a2="B"/>
<output a0="β'"/>
</command>
<element type="angle" label="β'">
<angleStyle val="1"/>
<value val="1.1170107212763711"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<show object="false" label="false" ev="4"/>
<condition showObject="β ≟ 90°"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="2"/>
</element>
</construction>
</geogebra>