<p>In \(\triangle ACD\), \(B\) and \(E\) are the midpoints of \(AC\) and \(AD\).</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(BE\parallel CD\qquad BE=\dfrac12CD\)</p></div></div>
<div class="graph-block" data-graph-config="eyJheGlzIjpmYWxzZSwiZ3JpZCI6ZmFsc2UsIndpZHRoIjozMDAsImhlaWdodCI6MjAwLCJlbGVtZW50cyI6W3siaWQiOiJpbXBvcnRlZC1wb2x5Z29uLWJsb2NrLTAtY29udGVudC1lbGVtZW50LTAiLCJ0eXBlIjoicG9seWdvbiIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1swLDBdLFsxLjM0LDEuNjddLFs1LjA0LDBdXSwiZmlsbENvbG9yIjoibm9uZSIsInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTEiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAuNjcsMC44MzVdLCJwb2ludDIiOlszLjE5LDAuODM1XSwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjoxLjE4LCJ5IjoxLjkyLCJpZCI6ImltcG9ydGVkLXRleHQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMiIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChBXFwpIiwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjowLjQsInkiOjAuOTIsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0zIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKEJcXCkiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOi0wLjI5LCJ5IjowLCJpZCI6ImltcG9ydGVkLXRleHQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtNCIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChDXFwpIiwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4Ijo1LjEsInkiOjAsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC01IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKERcXCkiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjMuMzEsInkiOjAuOTIsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC02IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKEVcXCkiLCJzdHJva2VXaWR0aCI6MS41fV0sImFsaWdubWVudCI6ImNlbnRlciIsInhBeGlzTGFiZWwiOiJcXCh4XFwpIiwieUF4aXNMYWJlbCI6IlxcKHlcXCkiLCJib3VuZGluZ0JveCI6Wy0wLjQ3MDY5OTk5OTk5OTk5OTksMi45ODQ3NjY2NjY2NjY2NjcsNS40Nzg1LC0wLjk4MTM2NjY2NjY2NjY2NjddLCJvcmlnaW5MYWJlbCI6IlxcKE9cXCkiLCJncmlkU2V0dGluZ3MiOnsibWFqb3JDb2xvciI6IiM5Y2EzYWYiLCJtYWpvclN0ZXBYIjoxLCJtYWpvclN0ZXBZIjoxLCJtaW5vckNvbG9yIjoiI2QxZDVkYiIsIm1ham9yT3BhY2l0eSI6MC40NSwibWlub3JPcGFjaXR5IjowLjQ1LCJtaW5vckVsZW1lbnRzWCI6NCwibWlub3JFbGVtZW50c1kiOjQsIm1ham9yU3Ryb2tlV2lkdGgiOjAuOCwibWlub3JTdHJva2VXaWR0aCI6MC41LCJzaG93TWFqb3JHcmlkTGluZXMiOnRydWUsInNob3dNaW5vckdyaWRMaW5lcyI6dHJ1ZX0sInNob3dBeGlzTGFiZWxzIjpmYWxzZSwic2hvd05hdmlnYXRpb24iOmZhbHNlLCJrZWVwQXNwZWN0UmF0aW8iOnRydWUsInhUaWNrTGFiZWxFdmVyeSI6MSwieVRpY2tMYWJlbEV2ZXJ5IjoxLCJ4QXhpc0xhYmVsT2Zmc2V0IjpbMiwtMTFdLCJ5QXhpc0xhYmVsT2Zmc2V0IjpbLTE0LDVdLCJvcmlnaW5MYWJlbE9mZnNldCI6Wy01LC0yXX0=" data-width="300" data-height="200" data-proportional style="width:300px;height:200px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:300px;--graph-aspect:300 / 200;"></div>
<p class="paragraph-header">Proof</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" data-proportional style="width:300px;height:200px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:300px;--graph-aspect:300 / 200;"></div>
<p>The smaller triangle is similar to the whole triangle with scale factor 1 : 2.</p>
<p class="paragraph-header">Equivalent labelling</p>
<p>If&nbsp;\(M,N\) are the midpoints of \(AB,AC\), then&nbsp;\(MN\parallel BC\) and \(MN=\frac12BC\).</p>

In \(\triangle ACD\), \(B\) and \(E\) are the midpoints of \(AC\) and \(AD\).

The smaller triangle is similar to the whole triangle with scale factor 1 : 2.

If \(M,N\) are the midpoints of \(AB,AC\), then \(MN\parallel BC\) and \(MN=\frac12BC\).

Midpoint Theorem

Midpoint Theorem

Video Solution:

Solution:

Solution locked