The coordinates of the feet of the perpendiculars from the vertices of a triangle to the opposite sides are $(20, 25)$,$(8, 16)$,and $(8, 9)$. Find the orthocenter of the triangle.

If vertices of a quadrilateral are $A(0,0), B(3,4), C(7,7)$ and $D(4,3)$,then the quadrilateral $ABCD$ is

In a $\triangle ABC$,$2x+3y+1=0$ and $x+2y-12=0$ are the perpendicular bisectors of its sides $AB$ and $AC$ respectively. If $A$ is $(3,2)$,then the slope of the side $BC$ is

If the three lines $x - 3y = p$,$ax + 2y = q$,and $ax + y = r$ form a right-angled triangle,then:

Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$,respectively. $A$ right-angled triangle $AMN$ is inscribed in the triangle $OAB$,where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$,respectively. If the area of the triangle $AMN$ is $\frac{4}{9}$ of the area of the triangle $OAB$ and $AN : NB = \lambda : 1$,then the sum of all possible values of $\lambda$ is:

The points $A(2a, 4a)$,$B(2a, 6a)$,and $C(2a + \sqrt{3}a, 5a)$,where $a > 0$,are the vertices of: