The orthocentre of the triangle formed by the lines $xy = 0$ and $x + y = 1$ is

If $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$ are the vertices of a triangle with side lengths $a, b, c$ opposite to vertices $A, B, C$ respectively,then the excentre with respect to vertex $B$ is:

Let $A(6,8)$,$B(10 \cos \alpha, -10 \sin \alpha)$ and $C(-10 \sin \alpha, 10 \cos \alpha)$ be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ are its orthocenter and centroid respectively,then $(5a - 3h + 6k + 100 \sin 2\alpha)$ is equal to . . . . . .

The orthocentre of the triangle with vertices $(0, 0)$,$(3, 4)$,and $(4, 0)$ is

Let the orthocentre and centroid of a triangle be $A(-3, 5)$ and $B(3, 3)$ respectively. If $C$ is the circumcentre of this triangle,then the radius of the circle having line segment $AC$ as diameter is:

Consider a triangle having vertices $A(-2, 3)$,$B(1, 9)$,and $C(3, 8)$. If a line $L$ passing through the circum-center of triangle $ABC$ bisects line $BC$ and intersects the $y$-axis at point $\left(0, \frac{\alpha}{2}\right)$,then the value of the real number $\alpha$ is $.....$