in an equilateral triangle prove that the centroid and circumcentre of the triangle coincide
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GIVEN: An equilateral triangle ABC, Medians AP, BQ, &CR. Their point of concurrency is O, which is the centroid of the triangle.
TO PROVE THAT: Centroid O is the circumcentre of the triangle ABC.
If we prove that centroid O is the circumcentre of the triangle, then it automatically becomes the centre of the circumcircle.
PROOF: Since AP is median, so P is mid point of BC.
ie, BP = PC.
AB = AC ( as triangle ABC is equilateral)
AP=AP ( common side)
Hence triangle ABP is congruent to ACP( by SSS congruence criterion)
=> angle APB = angle angleAPC ( corresponding parts of congruent triangles)
But their sum = 180°
So, each angle has to be 90°.
That shows that AP is perpendicular bisector of BC.
Similarly, pyove that BQ & CR are perpendicular bisectors of AC & AB respectively.
So now, The point of concurrency ‘O' of these perpendicular bisectors becomes circumcentre of the triangle. ( as circumcentre is the point of concurrency of 3 perpendicular bisectors of the sides of the triangle). And this centre is also the centre of circum circle.
This way centroid O coincides with circumcentre O…
[Hence Proved]
HOMEWORK HELP > MATH
In an equilateral triangle, prove that the centroid & the circumcentre of the triangle coincide.
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Expert Answers
VALENTIN68 | CERTIFIED EDUCATOR
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)