Browsing category geometry

Can you find a centre and angle of rotation without any construction?

Some time ago, I had a message from someone who - somewhat oddly - wanted to find a centre of rotation (with an unknown angle) without constructing any bisectors. (Obviously, if it was a right-angle rotation, they could use the set-square trick; if it was a half-turn, the centre of

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A Varignon Vector Masterclass

I recently listened to @mrhonner's episode of @myfavethm, in which he cited Varignon's Theorem as his favourite. What's Varignon's Theorem when it's at home? It states that, if you draw any quadrilateral, then connect the midpoints of adjacent sides, you get a parallelogram. Don't believe it? Try Mark's nifty geometry

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$\cos(72º)$, revisited again: De Moivre’s Theorem

In previous articles, I've looked at how to find $\cos(72º)$ using some nasty algebra and some comparatively nice geometry. In this one, inspired by @ImMisterAl, I try some nicer - although quite literally complex - geometry. De Moivre's Theorem I'm going to assume you're ok with complex numbers. If you're

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Ask Uncle Colin: A Troublesome Triangle

Dear Uncle Colin, I couldn't make head nor tail of this geometry problem: "If $a:b=12:7$, $c=3$, and $B\hat{A}C = 2 B\hat{C}A$, find the length of the sides $a$ and $b$." - Totally Rubbish In Geometry Hi, TRIG, and thank you for your message! (And don't put yourself down like that,

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Another @solvemymaths problem

Another geometry puzzle from @solvemymaths: I enjoyed this one -- no solution immediately jumped out at me, and I spend a great deal of time looking smugly at a way over-engineered circle theorems approach I can no longer remember. Let's label the apex of the triangle P, and the octagons

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A RITANGLE problem

When RITANGLE advises you to use technology to answer a question, you know it's going to get messy. So, with some trepidation, here goes: (As usual, everything below the line may contain spoilers.) It's easy enough to do this in Geogebra - but somehow a little bit unsatisfactory to move

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Daylight, Durlston Castle, and Where is Hamburg?

"Is Hamburg that much further north than London?" I furrowed my brow. Hamburg, to the best of my knowledge, is not that much further north than London. But here it was, written in stone (on the side of Durlston Castle in Swanage.) (I've transcribed the sign at the bottom of

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An “Impossible” New Zealand exam: Part I

To-may-to / tomato; potato / po-tah-to; impossible exam / underprepared students. This time it's the hapless Kiwis who are making Downfall parody videos and complaining that their practice papers hadn't prepared them for stuff on the syllabus. Never mind; the formidable @solvemymaths has picked out the two most-complained-about questions, and

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Another of Colin’s blasted puzzles

Before I begin: this post involves a puzzle and my attempt at a solution; everything above the horizontal rule is spoiler-free, but go beyond that at your peril. Some days, you can almost hear @colinthemathmo's chuckle as he innocently poses a question such as: Find all configurations of 4 points

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Tessellations and cuboids

On a recent1 episode of Wrong, But Useful, Dave mentioned something interesting2: if you take three regular shapes that meet neatly at a point - for example, three hexagons, or a square and two octagons - and make a cuboid whose edges are in the same ratio as the number

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