§ 3

In a Cartesian rectangular coordinate system the coordinates of the vertices of a triangle are given: $A(20,-15), B(-16,0)$ and $C(-8,6)$. Find the coordinates of the center and the radius of the circle inscribed in the triangle.

The origin lies in one of the angles formed by the lines with the equations $A_{1} x+B_{1} y+C_{1}=0$ and $A_{2} x+B_{2} y+C_{2}=0$ in a Cartesian rectangular coordinate system. Under what necessary and sufficient condition on the coefficients of the equations is this angle acute?

Form the equation of the line passing through the origin and meeting the lines with the equations $x=1+2 t$, $y=2+3 t, z=-t$ and $x=4 t, y=5-5 t, z=3+2 t$.

Find the radius and the coordinates of the center of the sphere passing through the point $A(0,1,0)$ and tangent to the planes with the equations

The coordinate system is Cartesian rectangular.

In a Cartesian rectangular coordinate system the coordinates of the vertices of a triangle $A(1,2,3), B(1,5,-1)$ and $C(5,3,-5)$ are given. Find the coordinates of the center of the circle circumscribed about the triangle.

Write the equations of the line which is parallel to the line $\mathbf{r}=\mathbf{r}_{0}+\mathbf{a}_{0} t$ and meets the lines $\mathbf{r}=\mathbf{r}_{1}+\mathbf{a}_{1} t$ and $\mathbf{r}=\mathbf{r}_{2}+\mathbf{a}_{2} t$. It is given that the vectors $\mathbf{a}_{0}, \mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are not coplanar.

Write the parametric equations of the line which is the intersection of the planes with the equations $\mathbf{r}=\mathbf{r}_{1}+u \mathbf{a}+v \mathbf{b}$ and $\left(\mathbf{r}-\mathbf{r}_{2}, \mathbf{n}\right)=0$.