Browsing category triangles

Ask Uncle Colin: Bridges, Donkeys and Triangles

Dear Uncle Colin, I'm struggling to understand why, if you know a triangle has two sides the same, the base angles must be the same. Can you explain? -- I'm Struggling Over Some Coherent Explanation Leveraging Equal Sides Hi, ISOSCELES, and thanks for your message! There are several good proofs

Fractions that generate Pythagorean Triples

An interesting tweet, some time ago, from @RJS2212: Two unit fractions where denominators differ by 2Add fractionsNumerator & denominator of the sum are two smaller numbers of Pythag triple — Robert J Smith (@RJS2212) May 30, 2016 And of course, you wonder two things: a) why does it work, and

An alternative proof of the $\sin(2x)$ identity

Uncle Colin recently explained how he would prove the identity $\sin(2x) \equiv 2 \sin(x)\cos(x)$. Naturally, that isn't the only proof. @traumath pointed me at an especially elegant one involving the unit circle. Suppose we have an isosceles triangle set up like this: The vertical 'base' of the triangle is $2\sin(\alpha)$

A MathsJam Masterclass: A Circle That Won’t Behave

Somewhere deep in the recesses of my email folder lurks a puzzle that looks simple enough, but that several of my so-inclined friends haven't found easy: A circle of radius $r$, has centre $C\ (0,r)$. A tangent to the circle touches the axes at $A\ (9,0)$ and $B\ (0, 2r+3)$.

Is there a tangent rule?

There's a natural question, when you learn about the sine and cosine rules: "Is there a tan rule?" The answer to that is yes - yes, there is a tan rule. The natural follow-up is "Why don't we learn it?" Let me explain why not! Here's the tangent rule in

Is there a tangent rule?

There's a natural question, when you learn about the sine and cosine rules: "Is there a tan rule?" The answer to that is yes - yes, there is a tan rule. The natural follow-up is "Why don't we learn it?" Let me explain why not! Here's the tangent rule in

Is there a tangent rule?

There's a natural question, when you learn about the sine and cosine rules: "Is there a tan rule?" The answer to that is yes - yes, there is a tan rule. The natural follow-up is "Why don't we learn it?" Let me explain why not! Here's the tangent rule in

Heroic triangles

In a recent Maths Challenge, students were told the area of a triangle ($7$cm$^2$) and the length of two of its sides ($6$cm and $8$cm), and asked how many possible lengths there were for the third side. It's easy enough to show there are two: let the base of the

Pascal’s tetrahedron

So, there I was, idly figuring out one of Barney's fiendish puzzles ("How many pairs of dice would you have to throw to be 95% certain of seven being the modal total?") when I started thinking about the binomial expansion (don't tell the Mathematical Ninja!) You know it: if you

The Secrets of the Mathematical Ninja: The World's Third Most Famous Triangle

My favourite trick, when I was helping students at the Physics Homework Centre at Montana State University, was to eyeball a question for a moment and say "... which is, what, 53.13 degrees or so..." without batting an eyelid. The poor students! There they were trying to figure out which