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

IP 3

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 $ABC$.

Solution:

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Answer:(i) $AD=DB$; all three perpendicular bisectors pass through $O$. (ii) Centre $O$, radius $OA=OB=OC$.

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