<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/1775569404482-iup7f5otrgj.png" style="max-width: min(280px, 100%); height: auto; display: block;" alt="Incentre diagram of triangle ABC" class="aligncenter custom-width" /></figure></div>
<p>In triangle \(ABC\), \(AO\) and \(BO\) bisects angles \(BAC\) and \(ABC\), respectively. \(AO\) and \(BO\) intersect at \(O\).</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-content"><span class="list-item-text">Prove the concurrency (intersecting at one point) of the \(3\) angle bisectors of a triangle by proving that&nbsp;\(CO\) bisects angle \(ACB\).</span></div></li><li data-list-item-id="item-ii"><div class="list-item-content"><span class="list-item-text">Hence prove that there exists a circle with centre \(O\) that touches \(AB\), \(BC\) and \(AC\) at one point, respectively.</span></div></li></ol>
<p><strong>Note:</strong> The point \(O\) is known as the <strong>incentre</strong> of triangle \(ABC\) and \(O\) is equidistant to the \(3\) sides of triangle \(ABC\).</p>

In triangle \(ABC\), \(AO\) and \(BO\) bisects angles \(BAC\) and \(ABC\), respectively. \(AO\) and \(BO\) intersect at \(O\).

Prove the concurrency (intersecting at one point) of the \(3\) angle bisectors of a triangle by proving that \(CO\) bisects angle \(ACB\).

Hence prove that there exists a circle with centre \(O\) that touches \(AB\), \(BC\) and \(AC\) at one point, respectively.

**Note:** The point \(O\) is known as the **incentre** of triangle \(ABC\) and \(O\) is equidistant to the \(3\) sides of triangle \(ABC\).

Concurrency of the Angle Bisectors of Triangles and the Incentre of Triangles

Concurrency of the Angle Bisectors of Triangles and the Incentre of Triangles