<p>In triangle \(ABC\), \(D\), \(E\) and \(F\) are the midpoints of \(AB\), \(BC\) and \(AC\), respectively. \(AE\) and \(BF\) are known as the <strong>medians</strong> of triangle \(ABC\). \(DG\) and \(FH\) are straight lines that pass through \(D\) and \(F\), respectively, and are parallel to \(AE\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1775801228745-c9kym7gfb8p.png" style="max-width: min(270px, 100%); height: auto; display: block;" alt="Triangle ABC with medians and parallel lines" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-content"><span class="list-item-text">Prove that&nbsp;\(BG = GE = EH = HC\).</span></div></li><li data-list-item-id="item-ii"><div class="list-item-content"><span class="list-item-text">Let&nbsp;\(O\) be the point of intersection of the medians \(AE\) and \(BF\). Hence or otherwise, prove that&nbsp;\(BO = 2OF\) and \(AO = 2OE\).</span></div></li><li data-list-item-id="item-iii"><div class="list-item-content"><span class="list-item-text">Use the fact that the result in <strong>(b)</strong> is valid for the point of intersection of any two medians of triangle \(ABC\) to prove the concurrency (intersecting at one point) of the \(3\) medians of a triangle, where&nbsp;\(O\) is the point of intersection of the \(3\) medians.</span></div></li></ol>
<p><strong>Note:</strong> The point \(O\) is known as the <strong>centroid</strong> of triangle \(ABC\) and \(O\) is the common point of trisection of the \(3\) medians of triangle \(ABC\).</p>
<p>There are many other interesting properties of the centroid of a triangle, which will not be presented here due to limited space. Readers may explore by themselves.</p>

In triangle \(ABC\), \(D\), \(E\) and \(F\) are the midpoints of \(AB\), \(BC\) and \(AC\), respectively. \(AE\) and \(BF\) are known as the **medians** of triangle \(ABC\). \(DG\) and \(FH\) are straight lines that pass through \(D\) and \(F\), respectively, and are parallel to \(AE\).

Let \(O\) be the point of intersection of the medians \(AE\) and \(BF\). Hence or otherwise, prove that \(BO = 2OF\) and \(AO = 2OE\).

Use the fact that the result in **(b)** is valid for the point of intersection of any two medians of triangle \(ABC\) to prove the concurrency (intersecting at one point) of the \(3\) medians of a triangle, where \(O\) is the point of intersection of the \(3\) medians.

**Note:** The point \(O\) is known as the **centroid** of triangle \(ABC\) and \(O\) is the common point of trisection of the \(3\) medians of triangle \(ABC\).

There are many other interesting properties of the centroid of a triangle, which will not be presented here due to limited space. Readers may explore by themselves.

Concurrency of the Medians of Triangles and the Centroid of Triangles

Concurrency of the Medians of Triangles and the Centroid of Triangles