Q. 33.7( 6 Votes )

# Let us prove that in an equilateral triangle, circumcentre, incentre, centriod, orthocenter will coincide.

Answer :

We know,

Circumcenter of a triangle: The intersection of perpendicular bisectors of the triangles.

Incenter: The intersection of angle bisectors of the triangle.

Centroid: The intersection of the medians of the triangle.

Orthocentr: The intersection of the altitudes of the triangle

In order to prove that, In an equilateral triangle, circumcenter, incentre, centroid and orthocenter coincide, it is sufficient to prove that for any side median, altitude, perpendicular bisector and angle bisector of angle opposite to that side is common.

Let us consider an equilateral ΔABC, such that AD is a median to side BC.

To Prove:

(i) AD ⊥ BC [AD is Altitude and perpendicular bisector of BC]

(ii) ∠BAD= ∠CAD [AD is angle bisector of ∠A]

Proof:

In ΔABD and ΔACD

AB = AC [Sides of equilateral triangle]

AD = AD [Common side]

BD = CD [As, AD is median to BC]

⇒ ΔABD ≅ ΔACD [By SSS property of congruent triangles]

Now, As corresponding parts of congruent triangles are equal [CPCT], we have

∠ADB = ∠ADC

Also,

∠ADB + ∠ADC = 180° [Linear pair]

⇒ ∠ADB + ∠ADB = 180°

⇒ ∠ADB = ∠ADC = 90°

⇒ AD ⊥ BC

Since, AD ⊥ BC, and BD = BC

∴ AD is perpendicular bisector of BC and as well as altitude from A to BC.

Now,

∠BAD = ∠CAD [By CPCT]

AD is angle bisector of ∠A.

∴ AD is median, perpendicular bisector, altitude and angle- bisector.

We can prove this result for every median, and since lines are same, their intersection will also be same.

Hence Proved!

Rate this question :

AD is the median and G is the centriod of an equilateral triangle. If the length of side 3√3 m, then let us write the length of AG.

West Bengal MathematicsLet us write what is the length of circum radius of a triangle lengths of sides of which are 6 cm, 8 cm and 10 cm.

West Bengal MathematicsIf the length of circum radius of a right-angled triangle is 10 cm, then let us write how much length of hypotenuse of triangle is.

West Bengal MathematicsI draw an obtuse angled triangle and let us prove that the perpendicular bisectors of sides are concurrent. Let us write where the circumcentre lies (inside/outside/on the side) of the triangular region.

West Bengal MathematicsRita draws a right angled triangle. Let us prove logically that the perpendicular bisectors of sides are concurrent and where the position of the circum centre (inside/outside/on the side) is

West Bengal MathematicsDEF is a pedal triangle of an equilateral triangle ABC. Let us write the measure of ∠FDA

West Bengal MathematicsIf the length of circumradius of a right angled triangle is 5 cm, then the length of hypotenuse is

West Bengal MathematicsAD, BE and CF are the medians of ΔABC. If AD = BC, then let us prove that the angle between two medians is 90°.

West Bengal Mathematics