Metric Problems Involving a Line: Perpendiculars, Angles, and Distances

A line $x-2 y+1 = 0$, containing the base $B C$ of a triangle $A B C$, and the vertex $A(3,-4)$ opposite this base, are given. Find the equation of the altitude dropped from $A$ to the side $B C$.

Find the projection of the point $(-3,7)$ onto the line $5 x-4 y+2 = 0$.

Find the point symmetric to the point $(0,11)$ with respect to the line $x+7 y-52 = 0$.

On the line $2 x-3 y+18 = 0$ find a point $C$ such that the sum of the distances to the two given points $A(-7,-3)$ and $B(5,5)$ is smallest.

A vertex of a triangle is at the point $(1,3)$, and the bisectors of two of its angles belong to the lines $y+3 = 0$ and $2 x+y+5 = 0$. Write the equation of the side of the triangle opposite this vertex.

Write the equations of the sides of an isosceles trapezoid, knowing the midpoints of its bases $(1,1),(2,8)$ and points $(4,-3),(-15,14)$ on its lateral sides.

The equation of a side of a rhombus, $x+3 y-8 = 0$, and the equation of its diagonal, $2 x+y+4 = 0$, are given. Write the equations of the remaining sides of the rhombus, knowing that the point $(-9,-1)$ lies on the side parallel to the given one.

On the line $2 x-3 y+6 = 0$ find the points at a distance $2 / 5$ from the line $3 x-4 y+11 = 0$.

Set up the equations of the lines parallel to the line $8 x + 6 y-7 = 0$ and situated at a distance 2 from it.

Through the point $(3,-1)$ draw a line at a distance 1 from the point $(2,-3)$.

Through the point of intersection of the lines $3 x+y+10 = 0,4 x+5 y + 6 = 0$, draw a line at a distance 4 from the origin.

Set up the equation of the locus of points equidistant from the lines $7 x+y+10 = 0$ and $x + y+2 = 0$.

Two intersecting lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x + B_{2} y+C_{2} = 0$, and a point $\left(x_{0}, y_{0}\right)$ not belonging to either of these lines, are given. Write the equation of the bisector of the angle between the lines in which this point lies.

Write the equation of the bisector of the angle between the lines $2 x-y-1 = 0$ and $11 x+2 y+8 = 0$, inside which the point $(1,2)$ lies.

Write the equation of the bisector of the acute angle between the two lines $x-3 y-1 = 0$ and $3 x-y+5 = 0$.

Write the equations of the tangents to the circle with center $(1,1)$ and radius 2, drawn from the point $(-7,-1)$.

Set up the equations of the bisectors of the interior angles of a triangle whose sides are given by the equations: $3 x-4 y = 0,4 x-3 y = 0,5 x+12 y-10 = 0$.

A vertex $(4,0)$ of a triangle and the altitudes drawn from its two other vertices, $3 x-y-2 = 0, x+y = 0$, are given. Find these vertices.

The center of symmetry of a square is at the point $(-1,0)$, and one of its sides is given by the equation $x+3 y-5 = 0$. Set up the equations of the three other sides.

Set up the equations of the sides of a square, two parallel sides of which pass through the points $(2,1)$ and $(3,5)$, and the other two through the points $(0,1)$ and $(-3,-1)$.

Set up the equation of the circle of radius $\sqrt{5}$, tangent to the line $x+2 y-3 = 0$, knowing that its center lies on the axis $O y$.

Set up the equations of the tangents to the circle

parallel to the line $3 x-4 y = 0$.

Set up the equations of the tangents to the circle

perpendicular to the line $4 x+3 y = 0$.

Find the center of the circle inscribed in the triangle bounded by the coordinate axes and the line $3 x-4 y-5 = 0$.

Find the center $C$ and radius $r$ of the circle passing through the point $(-1,3)$ and tangent to the lines $7 x+y = 0, x-y+8 = 0$.

Find the center $C$ and radius $r$ of the circle inscribed in the triangle whose sides are given by the equations: $x+y+12 = 0,7 x+y = 0$, $7 x-y+28 = 0$.

Find the coordinates of the center of the circle:

A triangle with vertices $A(1,1), B(-1,-1), C(7,-5)$ is given. Find the equations of the bisectors of the interior angles, the center $C$ of the inscribed circle, and its radius $r$.

Under the conditions of the preceding problem, find the center $C^{\prime }$ and radius $r^{\prime }$ of the circle tangent to the side $A B$ and to the extensions of the sides $A C$ and $B C$ of the triangle $A B C$.

Problem 321: a triangle with vertices $A(1,1), B(-1,-1), C(7,-5)$ is given.

Given two vertices $A(0,1), B(1,1)$ and the orthocenter $P(-3,4)$ of a triangle, find the third vertex $C$.

(Euler line.) Prove that in every triangle the orthocenter, the centroid, and the center of the circumscribed circle lie on one line.

Write the equations of the sides of a rhombus $A B C D$, knowing the point $M(1,6)$ where its diagonals meet, and points on three of its sides: $P(3,0)$ on $A B, Q(6,6)$ on $B C, R(5,9)$ on $C D$.

Write the equations of the sides of a square $A B C D$, knowing its center $(1,6)$ and points on two of its adjacent sides: $(4,9)$ on $A B,(-5,4)$ on $B C$.

A vertex $P(3,5)$ of an isosceles triangle, the equation of its base $x-2 y+12 = 0$, and the area $S = 15$ are given. Write the equations of the lateral sides.

Let $v_{1}$ and $v_{2}$ be direction vectors of the lines $\ell_{1}$ and $\ell_{2}$, respectively. It is known that the angle from the vector $v_{1}$ to the vector $v_{2}$ equals $6 \pi / 5$. What is the angle:

Determine the (signed) angle from the first line to the second, if their slopes $k_{1} = 1 / 3$ and $k_{2} = -1 / 2$, respectively, are known.

Through the point $(3,1)$ draw lines forming an angle $\pi / 4$ with the line $2 x+3 y-1 = 0$.

Find the interior angles of the triangle formed by the lines

The coordinates $\left(\alpha_{1}, \beta_{1}\right),\left(\alpha_{2}, \beta_{2}\right)$ of direction vectors of two intersecting lines are given. Express in terms of them the angle from the first line to the second.

Two points $A(3,3), B(0,2)$ are given. On the line $x+y-4 = 0$ find a point $M$ from which the segment $A B$ is seen at an angle $\pi / 4$.

The equations of the base of an isosceles triangle, $x+y-1 = 0$, and of one of its lateral sides, $x-2 y-2 = 0$, are given; the point $(-2,0)$ lies on the other lateral side. Find the equation of the third side of the triangle.

Two lines are given: $x+y = 0$ and $x-2 y+6 = 0$. Find a third line such that the second of the given lines is the bisector of the angle between it and the first line.

Two vertices of a triangle $A B C: A(1,2), B(3,4)$ and the tangents of the interior angles at these vertices, $\operatorname {tg} A = -1 / 2, \operatorname {tg} B = 1 / 3$, are given.

Find the third vertex of the triangle, knowing that it lies on the same side of the line $A B$ as the origin.

Two lines $x+y+1 = 0$ and $7 x-y+7 = 0$ are given. Find the cosine of the angle between them in which the point $(1,2)$ lies.

Two lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x+B_{2} y+C_{2} = 0$ and a point $\left(x_{0}, y_{0}\right)$ not belonging to either of these lines are given. Find the cosine of the angle $\varphi$ between them in which the given point lies.

Three lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x+B_{2} y+C_{2} = 0$ and $A_{3} x + B_{3} y+C_{3} = 0$ form a triangle. Find the cosine of the interior angle of this triangle formed by the first and second lines.

Three lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x+B_{2} y+C_{2} = 0$ and $A_{3} x + B_{3} y+C_{3} = 0$ pass through one point. Find the necessary and sufficient condition for the third line to pass through the acute angle formed by the first and second lines.