Loci

Chords of the circle $x^{2}+y^{2}=2ax$ are drawn from the origin. Find the locus of the midpoints of these chords.

Find the equation of the locus of points whose distances to the lines $Ax+By+C=0$ and $A_{1}x+B_{1}y+C_{1}=0$ are in the ratio $m:n$.

A segment of length $a+b$ slides with its ends on the coordinate axes. Find the curve traced by the point $M$ dividing it into parts $a$ and $b$. (Leonardo da Vinci's ellipsograph.)

Two circles of radii $b$ and $a$ are drawn centered at the origin. A variable radius meets the inner one at the point $A$, and the outer one at $B$. From $A$ a line is drawn parallel to the axis $Ox$, and from $B$ a line parallel to the axis $Oy$, up to their mutual intersection at the point $M$. Find the locus of the point $M$.

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.)

Chords of the circle $x^{2}+y^{2}=2ax$ are drawn from the origin and extended to their intersection with the line $x=2a$. These extensions are then laid off on the same chords starting from the origin. Find the locus of the endpoints of the shifted extensions. (Cissoid of Diocles.)

An arbitrary line is drawn from the origin, meeting the circle $x^{2}+y^{2}=ay$ and the line $y=a$ at the points $A$ and $B$. From the point $A$ a line is drawn parallel to the axis $Ox$, and from the point $B$ a line parallel to the axis $Oy$. Find the locus of the point of intersection of these lines. (Witch of Agnesi.)

Find the locus of points the product of whose distances to two given points $(a,0)$ and $(-a,0)$ is a constant quantity equal to $b^{2}$. (Ovals of Cassini.)

Find the locus of points the product of whose distances to two given points equals the square of half the distance between them. (Lemniscate of Bernoulli.)

Find the locus of points between whose distances to two given points there holds the linear relation $r_{1}-ar_{2}=b$. (Ovals of Descartes.)

Lines are drawn through the point $(0,-a)$ meeting the axis $Ox$. On each of them, on both sides of the point of intersection with $Ox$, segments of length $h$ are laid off. Find the locus of the endpoints of these segments. (Conchoid of Nicomedes.)

From a point $A$ on a circle of diameter $a$, secants are drawn. On each of them, on both sides of the other point of intersection with the circle, segments of length $b$ are laid off. Find the locus of their endpoints. (Limaçon of Pascal.)

A thread wound around the circle $x^{2}+y^{2}=a^{2}$ is unwound while remaining taut. Find the locus traced by its end, if the initial position of the end of the thread was at the point $(a,0)$. (Involute of a circle.)

A circle of radius $a$ rolls without slipping along the axis $Ox$. Find the equation of the curve traced by the point of the circle that, at the initial moment, touched the axis $Ox$ at the origin. (Cycloid.)

Find the equation of the curve traced by a point of a circle of radius $a$ rolling without slipping along the axis $Ox$, if, at the moment when the circle touched the axis $Ox$ at the origin, this point occupied the position $(0,a-b)$. (Trochoid.)

A circle of radius $a$ rolls without slipping on the outside of the circle $x^{2}+y^{2}=a^{2}n^{2}$. Find the locus traced by the point of the rolling circle that, at the initial moment, touched the fixed circle on the axis $Ox$. (Epicycloid.)

A circle of radius $a$ rolls without slipping inside the circle $x^{2}+y^{2}=a^{2}n^{2}$. Find the locus traced by the point of the rolling circle that, at the initial moment, touched the fixed circle on the axis $Ox$. (Hypocycloid.)

Show that for $n=4$ the hypocycloid becomes the astroid $x^{2/3}+y^{2/3}=(4a)^{2/3}$.

For $n=1$ the epicycloid becomes a curve called the cardioid. Show that the equation of the cardioid, in suitably chosen polar coordinates, has the form $r = a(1+\cos \varphi )$.

Show that the cardioid (see the preceding problem) is a special case of the limaçon of Pascal (see problem 141).

Show that for $n=2$ the hypocycloid becomes a line. (Cardano's theorem.)

A right triangle with legs $a$ and $b$ slides with its ends on the coordinate axes. Find the locus of the vertex of the right angle.

Find the locus of the centers of rectangles inscribed in a given triangle so that one of their sides lies on the base of the triangle.

Find the locus of the centers of parallelograms inscribed in a given quadrilateral, whose sides are parallel to the diagonals of the quadrilateral.

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.

Find the locus of the centroids of the triangles formed by the lines $x=0$, $y=0$, $\frac{x}{k}+\frac{y}{3-k}=1$.

Three vertices of a parallelogram, the directions of whose sides are given, slide along three given lines. What is the locus of the fourth vertex?

The four sides of a changing parallelogram all pass, at all times, through four given points on a line. Show that the diagonals of the parallelogram also pass through certain fixed points.

The sides of a right angle, whose position changes, pass at all times through two given points. Prove that its bisector also passes through a fixed point.

A side of a variable square passes through the origin, and its endpoints slide along lines parallel to the axis $Ox$. What is the locus of the other two vertices?

Find the locus of the midpoints of the chords of the circle $x^{2}+y^{2}=a^{2}$ passing through the point $P(c,0)$ inside the circle.

Lines $a$, $b$ and $c$ rotate about the points $A$, $B$, $C$ lying on one line. As they do so, the point of intersection of the lines $a$ and $b$ slides along a line $P$, and the point of intersection of the lines $a$ and $c$ slides along a line $Q$. Prove that the locus of the points of intersection of the lines $b$ and $c$ is a line $R$ passing through the point of intersection of the lines $P$ and $Q$.

Prove Desargues's theorem: if the vertices of two triangles lie on three lines issuing from one point, then the three points of intersection of the corresponding pairs of sides of the triangles lie on one line.