Question

prove that the centroid and circumcentre of an equilateral triangle are coincident

Answer 1

Hi there☺️
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⏬Solution:⏬

• Given:
ABC is an equilateral triangle (see the attachment)

• To prove:
Centroid and circumcentre of ∆ABC are coincident.

• Proof:
=> Let G and O be the centroid circumcentre respective of ∆ABC. Let the medians AD, BE and CF pass through G.

=> In ∆ABD and ∆ACD,

=> AB = AC (Since ∆ABC is equilateral)

=> AD = AD

=> BD = DC (Since AD is a median)

=> Therefore, ∆ABD is congruent to ∆ACD (by SSS congruency rule)

=> Therefore, angle ADB = angle ADC

=> But angle ADB + angle ADC = 180°

=> Therefore, angle ADB = angle ADC = 90° ; therefore, AD is perpendicular to BC

=> Now D is the midpoint of BC and AD is perpendicular to BC,

=> So, AD is also perpendicular bisector of BC, but AD is also a median.

=> Thus, in an equilateral triangle, the median of any side is also perpendicular bisector of that side.

=> So, the point of intersection of the medians and that of perpendicular bisectors is the same point.

=> Hence, centroid and circumcentre of an equilateral triangle are coincident.

✴️.....HENCE PROVED.....✴️
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Hope it helps✌️