<p>In triangle \(ABC\), \(EO\) and \(FO\) are the perpendicular bisectors of sides \(BC\) and \(AC\), respectively. \(D\) is a point on \(AB\) such that&nbsp;\(DO\) is perpendicular to \(AB\).</p>
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<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 \(3\) perpendicular bisectors of the sides of a triangle by proving that&nbsp;\(D\) is the midpoint of \(AB\).</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 passes through \(A\), \(B\) and \(C\).</span></div></li></ol>
<p>Note: The point \(O\) is known as the <strong>circumcentre</strong> of triangle \(ABC\) and \(O\) is equidistant to the \(3\) vertices of triangle \(ABC\).</p>

In triangle \(ABC\), \(EO\) and \(FO\) are the perpendicular bisectors of sides \(BC\) and \(AC\), respectively. \(D\) is a point on \(AB\) such that \(DO\) is perpendicular to \(AB\).

Prove the concurrency (intersecting at one point) of the \(3\) perpendicular bisectors of the sides of a triangle by proving that \(D\) is the midpoint of \(AB\).

Hence prove that there exists a circle with centre \(O\) that passes through \(A\), \(B\) and \(C\).

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

Concurrency of the Perpendicular Bisectors of the sides of Triangles and the Circumcentre of Triangles

Concurrency of the Perpendicular Bisectors of the sides of Triangles and the Circumcentre of Triangles