Geometry

Circumradius of a $\triangle ABC$ is $2$, O is the circumcentre, H is the orthocentre then $\dfrac{1}{64}(AH^{2}+BC^{2})(BH^{2}+AC^{2})(CH^{2}+AB^{2})$ is equal to

If the axes are translated to the circumcentre of the triangle formed by $(9,3),(-1,7),(-1,3)$, then the centroid of the triangle in the new system is

In $\displaystyle \Delta ABC,AB$ is diameter of circumcircle. If H is orthocentre of $\displaystyle \Delta ABC,$ then AH divides BC in the ratio

In $\displaystyle\Delta ABC,AD$ is perpendicular to $BC$ and $O$ is circumcentre of $\displaystyle\Delta ABC$, $\displaystyle \angle B=55^{\circ}, \angle A=80^{\circ}.$ Find $\displaystyle \angle OCB$

Let ABC be an acute-angled triangle. The circle $\Gamma$ with BC as diameter intersects AB and AC again at P and Q, respectively. Determine $\angle$BAC given that the orthocentre of triangle APQ lies on $\Gamma$.

In a $\displaystyle \triangle ABC$, G is centroid Then area of $\displaystyle \triangle AGC$ is_______of area of $\displaystyle \triangle ABC$

Orthocentre of an acute triangle $ABC$ is at the origin and its circumcenter has the co-ordinates ( $\dfrac{1}{2}, - \dfrac{1}{2}$). If the base $BC$ has the equation $4x - 2y = 5$, then the radius of the circle circumscribing the $\triangle ABC$ is

Let $f$, $g$, $h$ be the lengths of the perpendiculars from the circumcentre of the $ \Delta ABC$ on the sides BC, CA and AB respectively. If $\displaystyle \frac{a}{f}+\frac{b}{g}+\frac{c}{h}=\lambda \frac{abc}{fgh},$ then the value of $\displaystyle "\lambda "$ is

Let $ABC$ be a triangle with $\angle A = 45^{\circ}$. Let $P$ be a point on the side $BC$ with $PB = 3$ and $PC = 5$. If $'O'$ is the circumcentre of the triangle $ABC$ then the length $OP$ is equal to

Given a circle with centre $O.$ The smallest chord $PQ$ is of length $4$ cm, largest chord $AB$ is of length $10$ cm, and chord $EF$ is of length $7$ cm. Then, the radius of the circle is