Articles of plane geometry

Locus of intersection of two perpendicular normals to an ellipse

I came up with the following question while playing around with Geogebra, and rather than do it myself I figured I’d offer it here. For a given ellipse, the director circle is defined to be the set of all points where two perpendicular tangents to the ellipse cross each other. Suppose we instead consider the […]

Why are the coefficients of the equation of a plane the normal vector of a plane?

Why are the coefficients of the equation of a plane the normal vector of a plane? I borrowed the below picture from Pauls Online Calculus 3 notes: And I think the explanation he provides is great, however, I don’t understand how one of the concepts work. If the equation of a plane is $ax+by+cz=d$ […]

How can the notion of “two curves just touching” (vs. “two curves intersecting”) be expressed for a given metric space?

A popular introductory description of a “tangent (in geometry)” is presented as “the tangent line (or simply tangent) to a plane curve at a given point is the straight line that “just touches” the curve at that point.“ I like to find out whether and how this description might be expressed or translated in the […]

Problem on circles, tangents and triangles

Let $c_1,c_2,c_3$ be three circles of unit radius touching each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\Delta ABC$. Find the length of each side of $\Delta ABC$. I don’t have a clue how […]

Uniqueness of a configuration of $7$ points in $\Bbb R^2$ such that, given any $3$, $2$ of them are $1$ unit apart

This question from earlier today asks (paraphrasing here): Is there a configuration of $7$ points in the Euclidean plane such that, given any $3$ of the $7$ points, at least $2$ of them are $1$ unit apart? One such configuration, given in this answer, is the set of blue points in this diagram; the red […]

Another chain of six circles

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane). This is a generalization of the last my previous question in Three chains of six circles. (Noting that, in this configuration: $P’_1P_1P_4P’_4$ don’t lie on a circle) I am looking for a proof of the conjecture […]

Seven points in the plane such that, among any three, two are a distance $1$ apart

Is there a set of seven points in the plane such that, among any three of these points, there are two, $P, R$, which are distance $1$ apart?

Three normals of a hyperbola passing through the same point on the curve

The normals at three points $P$, $Q$, $R$ on a rectangular hyperbola $xy = c^2$ intersect at a point on the curve. Prove that the centre of the hyperbola is the centroid of the triangle $PQR$. I copied the the question word by word. Let assume that the points are (c$ t_1$,$\frac{c}{ t_1}$) , (c$ […]

Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area

Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area.