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A little terminology
When discussing tangency, it is important to keep a few basic definitions and properties in mind, especially when tackling somewhat elaborate constructions…

The tangent at X: École Polytechnique slang | Tangente
The right to wear a sword, formerly reserved for sergeants, was not granted to students at the École polytechnique until after the three revolutionary days of July 1830.

Pitot's theorem for quadrilaterals | Tangente
First stated by the French engineer and physicist Henri Pitot (1695–1771), the theorem that bears his name concerns quadrilaterals with an inscribed circle (known as tangential quadrilaterals).

A theorem puzzle in pictures: Monday's challenge | Tangente
Since 2015, the Image des maths website (https://images.math.cnrs.fr) has featured a "picture-theorem puzzle" every Monday, taken from Geometry in Figures, a book published in 2011 by the young Russian mathematician Arseniy Akopyan (CreateSpace Independent Publishing Platform) and available in French as Figures sans paroles (Independently Published, 2019).

The beautiful geometry of sangaku
A sangaku was originally a Japanese wooden votive tablet. It sometimes bears an engraved geometric figure with the statement of a problem, together with the solution or sometimes a hint.

The origins of the tangent
“Tangent” is a word that has found its way into everyday language. But where did it come from?

Catriona Agg's geometry puzzles | Tangente
Catriona Agg (until recently also known as Catriona Shearer) is a British mathematics teacher from Cambridge.

Editorial
"Tangente", from the Latin tangere, "to touch". If it's the name of your favorite magazine, it is because the word is rich in meanings that have opened up entire branches of mathematics.
