Intereting Posts

How many different $4$ letter words can you create from the characters of “parallelogram”?
Defining a metric space
Every prime $p_{n}$ is a prime factor of $\frac{1}{2}$ of all the square-free numbers.
Do finite products commute with colimits in the category of spaces?
What do the eigenvectors of an adjacency matrix tell us?
What are the most important questions or areas of study in the philosophy of mathematics?
Describe all one-dimensional representations of the alternating group A4.
An almost Fresnel integral
Quotient of the ring of integers of a number field by a prime ideal
Show that $\{1, \sqrt{2}, \sqrt{3}\}$ is linearly independent over $\mathbb{Q}$.
Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$?
There are $12$ stations between A and B, in how many ways you can select 4 stations for a halt in such a way that no two stations are consecutive
Why do we use “congruent to” instead of equal to?
Finding the derivatives of inverse functions at given point of c
Rigorous Proof: Circle cannot be embedded into the the real line!

From googling, it seems commonly believed that Euclid did this, but it seems nowhere in Euclid does he even state this property of a tangent line explicitly. Rather Euclid gives 4 other equivalent properties, that the line does not cross the circle, that it is perpendicular to the radius, that is a limit of secant lines, and that it makes an angle of zero with the circle, the first of which is his definition, the others being in Proposition III.16. I am wondering where the “meets only once” definition got started. I presume once it got going, and people stopped reading Euclid, (which seems to have occurred over 100 years ago), the currently popular definition took over. Perhaps I should consult Legendre or Hadamard? Thank you for any leads.

Well I have found this definition in Hadamard’s lessons in plane geometry. Any earlier references?

I have also found another equivalent characterization of a tangent by Euclid, Prop. (III.36-37): A segment PX, from a point P outside a circle and meeting the circle at X, is tangent to the circle at X if and only if there exists another segment PB, meeting the circle first at A and then at B, such that (PA)(PB) = (PX)^2, [in terms either of equality squares of lengths of segments, or of equality of area of rectangles].

- Orthogonal projection of a point onto a line
- Area of a triangle in terms of areas of certain subtriangles
- Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications
- Find length of $CD$ where $\measuredangle BCA=120^\circ$ and $CD$ is the bisector of $\measuredangle BCA$ meeting $AB$ at $D$
- Intersection of Altitudes in Hexagon
- Show that if an ellipse and a hyperbola have the same foci, then at each point of intersection their tangent lines are perpendicular.

- What is a composition of two binary relations geometrically?
- Find the sum of angles without trigonometry?
- Drawing a tetrahedron from a parellelepiped to convince myself it is 1/6th the volume,
- Equation of Cone vs Elliptic Paraboloid
- Derive the centroid of an area from a limiting procedure
- Analytic versus Analytical Sets
- problem on intersecting circles
- Need help with this geometry problem on proving three points are collinear
- Unit circle is divided into $n$ equal pieces, what is the least value of the perimeters of the $n$ parts?
- Penrose's remark on impossible figures

Some googling points to Archimedes (use ctrl-F to find his name in this article)

- need help to understand the following expression
- Prove that no function exists such that…
- What methods are known to visualize patterns in the set of real roots of quadratic equations?
- Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic?
- A multiplication algorithm found in a book by Paul Erdős: how does it work?
- How to find Laurent series Expansion
- Evaluating $\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$
- $2$-dimensional Noetherian integrally closed domains are Cohen-Macaulay
- How do I prove this sum is not an integer
- What is wrong in my proof? (Uniform convergence and Lebesgue integral)
- Fourier-Series of a part-wise defined function?
- half space is not homeomorphic to euclidean space
- How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
- How to calculate the distance between this two houses?
- probability circle determined by chord determined by two random points is enclosed in bigger circle