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

From figure, AD, BE and CF are medians of the triangle. Let G be the centroid of triangle ABC. Triangle ABC is an equilateral triangle, ∴ AB = BC = CA and ∠ABC = ∠BAC = ∠BCA = 60° In △BFC and △BEC, ⇒ BC = BC (Common Side) ⇒ ∠FBC = ∠ECB = 60°. ⇒ BF = EC (As F is mid-point of AB and E is mid-point of AC and AB = AC.) △BFC ≅ △BEC (By SAS axiom.) ∴ BE = CF (By C.P.C.T.) .........(1) Now, in △ABE and △ABD AB = AB (Common Side) ∠BAE = ∠ABD = 60° BD = AE (As D is mid-point of BC and E is mid-point of AC and BC = AC.) △ABE ≅ △ABD (By SAS axiom.) ∴ BE = AD (By C.P.C.T.) ............. (2) From equation 1 and 2, we get: ⇒ AD = BE = CF ⇒ We know that G (the centroid) of the triangle divides the median in a 2 : 1 ratio. ∴ GA = GB = GC. So, we can say that G is equidistant from the three vertices A. B and C. G is circumcentre of ΔABC. Hence, proved that the centroid and circumcentre are coincident.

If a diameter of a circle is perpendicular to one of two parallel chords of the circle, prove that it is perpendicular to the other and bisects it.

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The line joining mid-points of two chords of a circle passes through its center. Prove that the chords are parallel.

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In the figure (i) given below, OD is perpendicular to the chord AB of a circle whose center is O. If BC is a diameter, show that CA = 2OD.

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In the figure (ii) given below, O is the center of a circle. If AB and AC are chords of the circle such that AB = AC and OP ⊥ AB, OQ ⊥ AC, prove that PB = QC.

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Mathematics

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

Circles

Related Questions

If a diameter of a circle is perpendicular to one of two parallel chords of the circle, prove that it is perpendicular to the other and bisects it.

The line joining mid-points of two chords of a circle passes through its center. Prove that the chords are parallel.

In the figure (i) given below, OD is perpendicular to the chord AB of a circle whose center is O. If BC is a diameter, show that CA = 2OD.

In the figure (ii) given below, O is the center of a circle. If AB and AC are chords of the circle such that AB = AC and OP ⊥ AB, OQ ⊥ AC, prove that PB = QC.