In an equilateral triangle, prove that the centroid & the circumcentre of the triangle coincide.

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valentin68 | College Teacher | (Level 3) Associate Educator

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See the attached figure.

(ABC are triangle vertices, DEF are midpoints on triagle sides and O is the intersection of medians)

In triangles ABD and ACD

BD= CD (by median definition)

AD= AD (common segment)

angles ABC= ACB (equlateral triangle)

results triangles ABD=ACD

results angle BAD=CAD (=60/2=30 degree)

By smilarity all angles BAD=CAD=CBE=BCF=ACF=BCF(=30 deg)

We know angle ABC=ACD=BAC (=60 deg) because the triangle is eqilateral

Results angles ADC=ADB=BEA=BEC=CFA=CFB (=90 deg)

Now the triagle is equilateral it means all medians are equal

AD=BE=CF

Now we demonstrate triangles OBD=OCD

OD=OD (common degment)

CD=BD (by definition of median)

angle ODC=ODB (=90 deg)

Results OB=OC and by similarity

OB=OC=OE=OA=OF

It means circumcenter = centroid (qvod erat demonstratum)

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aruv | High School Teacher | (Level 2) Valedictorian

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Let ABC be an equilateral triangle. Also let D,E and F are resp. mid points of side AB ,BC and AC.Join A to E,B to F and C to D and intersect these at G. G  is centroid of triangle ABC.Since triangle is an equilateral therefore AE=CD=BF.Also CD will perpendicular to AB ,AE perpendicular to BC and BF perpendicular to AC.

In triangle ODB and OEB.

angle ODB=angle OEB = 90  (Right  angle)

OB=OB           (common)

BD= BE         ( AB=BC ,(1/2)AB=(1/2)BC)

Thus triangle ODB is congruence to OEB by RHS.

So

OD=OE

similarly we can prove  OD=OF

Thus O is incentre of triangle ABC ,which centroid.

To prove O as circumcentre.

consider triangle OAD and triangle OBD

please add to proof.

AD = BD        ( D is mid point of AB)

angle ODA= angle ODB =90  (proved)

OD  = OD   (common)

Thus triangle OAD is congruence to triangle OBD  by SAS.

Therefore    OA = OB

Similarly  ,we can prove  OA = OC

Therefore  O  is circumcentre of triangle ABC.

Thus O is centroid ,incentre and circumcentre of triangle ABC.

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