Loci and curves traced by linkages

A line is drawn from the origin making an angle $\frac{\pi \theta }{2}$ with the axis $Oy$. This line meets the line $x=a\theta$ at the point $M$. Find the equation of the locus of the point $M$. (Quadratrix of Dinostratus.)

The endpoints of the base of a triangle are at the points $(\pm a,0)$. One of the angles at the base is twice the other. Find the locus of the vertex of the triangle.

The endpoints of the base of a triangle are at the points $(\pm a,0)$. The difference of the angles at the base equals $\varphi$. Find the locus of the vertex.

A flat eccentric disk is mounted on a horizontal axis. What shape must it have so that a vertical rod resting against it performs harmonic oscillatory motion when the axis rotates uniformly?

The same question, but the vertical rod must perform uniform motion up and down. (This is how the tube of certain microscopes moves.)

An obliquely cut cylinder is covered with coloring substance and rolls along a plane. What curve bounds the colored part of the plane?

A straight line of given length $l$ slides with its ends $A$ and $B$ along the axes $Ox$ and $Oy$. From the vertex $C$ of the rectangle $OACB$ a perpendicular is dropped onto the line. Find the locus of the foot of this perpendicular.

The same question for the endpoint of the perpendicular dropped from the origin. (The four-petaled rose.)

A segment $OO_{1} = a$ is given. From an arbitrary ray through the point $O$, a perpendicular $O_{1}M_{1}$ is dropped from the point $O_{1}$, and from the foot of this perpendicular $M_{1}$ a perpendicular $M_{1}M$ is dropped onto the ray symmetric to the first one with respect to the line $OO_{1}$. Find the locus of the feet $M$ of the second perpendicular.

From a point $M(\xi , \eta )$ on the circle $x^{2}+y^{2}=a^{2}$, a line $y=\eta$ is drawn to its intersection with the $y$-axis at a point $N$. From the point $(a, 0)$ a line is drawn to the point $N$. Find the locus of the point of its intersection with the ordinate of the point $M$.

A right angle rotates about its vertex $A(a, 0)$. From the origin, through the point where one side of the angle meets the circle $x^{2}+y^{2}=r^{2}$, a line is drawn to its intersection with the other side of the angle. Find the locus of the point of intersection.

A right triangle has legs $AC = a$ and $BC = 2a$. The point $B$ slides along the axis $Ox$, and the point $A$ slides along the circle $x^{2}+y^{2}=a^{2}$. Find the locus of the vertices of the right angle.

Find the locus of the points of tangency of lines from the origin that are tangent to circles of radius $a$ with centers on the axis $Ox$.

Two equal parabolas rotate about their vertices so that the four points of their intersection always lie on a circle. Find the locus of the center of the circle.

Find the locus of the vertex of an equilateral (rectangular) hyperbola with a given center, passing through a given point.

The same question for the focus of the hyperbola.

A hyperbola with semi-axis $a$ passes through two given points at distance $2c$ apart. Find the locus of the center of the hyperbola, if it is equilateral.

Find the locus of the midpoints of the segments of normals to the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ lying inside the ellipse.

Two rods of length $\sqrt{2}$ rotate about their fixed endpoints at the points $(1,0)$ and $(-1,0)$. Their other endpoints are joined by a hinge to a rod of length $2$. Determine what curve the midpoint of the connecting rod describes.

The Peaucellier–Lipkin inversor consists of a rhombus $ABCD$ with side equal to $a$. The points $A$ and $C$ are joined to a fixed point $O$ by rods $OA$ and $OC$ of length $b$, where $b > a$.

The sides of the rhombus and the segments $OA$ and $OC$ are rods joined by hinges. Prove that the points $O$, $B$, and $D$ lie on one line, and moreover $OB \cdot OD = b^{2}-a^{2}$, so that the motion of a vertex of the rhombus along a given curve allows one to carry out an inversion.

Prove that when the point $B$ in the Peaucellier–Lipkin inversor moves along a circle passing through the point $O$, the point $D$ describes a straight line.

Let $z_{1}$ and $z_{2}$ be fixed points of the complex plane. Give a geometric description of the set of all points $z$ satisfying the equation:

A circle of radius $a$ with center at a point $O$ and its diameter $B C$ are given. A variable point $A$ moves along the circle. On the ray $O A$ a segment $O M$ is laid off, equal to the distance from the point $A$ to the line $B C$. Find the curve traced by the point $M$.

A point $O$ and a line $\ell$ at a distance $a$ from $O$ are given. A ray rotates about the point $O$, intersecting the line $\ell$ at a variable point $P$. On this ray, a segment $O M$ is laid off from the point $O$ such that $\left|O M\right| \cdot \left|O P\right| = b^{2}$. Find the curve traced by the point $M$ as the ray rotates.

In a circle of radius $a$ a diameter $O A$ is drawn. A ray rotates about its endpoint $O$, intersecting the circle at a variable point $B$. On the extension of the chord $O B$ beyond the point $B$, a segment $B M$ equal to $A B$ is laid off. Find the curve traced by the point $M$ as the ray rotates.

Two vertices of a triangle are fixed at the points $A$ and $B$, with $\left|A B\right| = c$; the third vertex moves along a circle of radius $b$ centered at the point $A$. What curve is traced by the point $D$ where the bisector of the angle $A$ meets the side $B C$?

A segment of constant length $2 a$ slides with its endpoints along two mutually perpendicular lines. Find, in generalized polar and rectangular coordinates, the equation of the curve traced during this motion of the segment by the foot of the perpendicular dropped from the point of intersection of the lines onto the segment.

A segment of constant length moves so that one of its endpoints slides along the circle $x^{2}+y^{2} = r^{2}$, while the other slides along the axis $O x$ (a slider-crank mechanism). Set up the equation of the curve traced by the point of the segment that divides it into parts of lengths $a$ and $b$.

Three vertices of a rhombus lie on the sides $A B, B C$ and $C D$ of a given square, respectively. Where can the fourth vertex lie?

The vertex $O$ of a hinged parallelogram $O P M Q$ is fixed, and the sides $O P$ and $O Q$ rotate with equal angular speed in opposite directions. Along what curve does the vertex $M$ move?

Two lines in the plane rotate in opposite directions with equal angular speed about the points $A$ and $B$ lying on them. Along what curve does their point of intersection move?

A segment $A B$ is given in the plane. Find the locus of points $M$ such that $\angle M B A = 2 \angle M A B$.