If S ,O and G represents circumcentre orthocentre and centroid respectively in a triangle then show that OG:GS=2:1

In a triangle, if orthocentre, circumcentre, incentre and centroid coincide, then the triangle must be_____

Let ABC be a triangle having O and I as its circumcentre and incentre, respectively. If R and r are the circumradius and the inradius respectively, then prove that (IO) 2 =R 2 −2Rr. Further show that the triangle BIO is right angled triangle if and only if b is the arithmetic mean of a and c.

If the orthocentre and centroid of a triangle are (-3, 5) and (3, 3) then its circumcentre is :

Let O be the circumcentre and H be the orthocentre of an acute angled triangle ABC. If A gt B gt C , then show that Ar (Delta BOH) = Ar (Delta AOH) + Ar (Delta COH)

Statement 1: If in a triangle, orthocentre, circumcentre and centroid are rational points, then its vertices must also be rational points. Statement : 2 If the vertices of a triangle are rational points, then the centroid, circumcentre and orthocentre are also rational points.

O and C are respectively the orthocentre and circumcentre of an acute angle triangle PQR. The points P and Q are joined and produced to meet the side QR ats. If anglePQS = 60^@ and angleQCR =130^@ then angleRPS =

Prove that the circumcenter, orthocentre, incenter, and centroid of the triangle formed by the points A(-1,11),B(-9,-8), and C(15 ,-2) are collinear, without actually finding any of them.

The points P,Q and R represent the complex numbers (sqrt2+isqrt2),(sqrt3+i) and (1+isqrt3) respectively in the z-plane. Show that the triangle PQR is isoseceles.

The corodinates of the three vertices of a triangle are (a, a tan alpha) , (b ,b tan beta) and (c, c tangamma) . If the circumcentre and orthocentre of the triangle are at (0,0) and (h,k) respectively, prove that (h)/(k)=(cos alpha+ cos beta+ cos gamma)/( sin alpha+ sin beta + sin gamma) .

P is the variable point on the circle with center at C. C A and C B are perpendiculars from C on the x- and the y-axis, respectively. Show that the locus of the centroid of triangle P A B is a circle with center at the centroid of triangle C A B and radius equal to the one-third of the radius of the given circle.