Posted in ask uncle colin, geometry, 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

Read More →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

Read More →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 $2\sin(\alpha)$

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Posted in circles, geometry, triangles, trigonometry.

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 $B\ (0, 2r+3)$.

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Posted in gcse, triangles, trigonometry.

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

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Posted in gcse, triangles, trigonometry.

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

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Posted in gcse, triangles, trigonometry.

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

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Posted in geometry, triangles, trigonometry.

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

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Posted in triangles.

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

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Posted in ninja maths, radians, triangles.

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

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