This sounded like a portent of doom

“So fear the wren”.

But the speaker was actually talking about a film star.

Interesting how cliches morph into phrases totally disconnected with the original. Tonight I heard ‘Beats into a cocktail’. The speaker didn’t mean egg flip, just that A was hugely better than B.

But that’s how language changes and all power to it.

It’s hot now and we complain.

But soon the snow will blow.

At the first flake, we’ll shiver and shake.

Wishing summer back again.

Problem 1358 at gogeometry [1] asked for a proof that in a regular 12-sided polygon the four diagonal shown all meet at a point. This is quite surprising; it\'s not hard to find threee diagonals the meet at a single point but four is rarer.

After playing arouind for a while I found a proof which was reasonably neat, but I had to use sines and cosines at one point, and I'd hoped for something simpler; and there was nothing very illuminating about my proof in any case. Stan Fulger came up with something much more insightful. Here is his beautiful answer.

It uses two well-known facts about triangles.

The altitudes of a triangle, i.e. the lines drawn from each vertex at 90° to the opposite side, meet at a point.

The angle bisectors of a triangle, i.e. the lines which divide each angle in half, meet at a point.

For example

Now for a "look and see" proof.

In the picture below three diagonals are angle bisectors in the blue triangle, so they meet at a point. Also three diagonals are altitudes of the orange triangle and therefore meet at a point. Two of the diagonals are both a bisector in one triangle and an altitude in the other. Therefore all four diagonals meet at a point.

I drew the figures above using GeoGebra classic.

[1] Geometry problem 1358 https://gogeometry.com/school-college/4/p1358-dodecagon-regular-concurrency-diagonal-infographic-classes.htm

I bought some drops for removing earwax. They contain Urea Hydrogen Peroxide...

So you are literally putting urea in your ear.

Trivial yet extraordinary subject of theory (10) 

How do you persuade a profiterole to go away? Say choux.

lachsschinken: A kind of cured, salted, smoked ham originally from Bavaria. The name derives from the German lachs  + schinken = ‘salmon ham’, from the bright pink colour of the meat. I found the word from today’s Word watch in The Times and what impressed me was the consonant cluster ‘chssch’ in the middle.

I went into a marquee and hung about a bit. A bloke came up and asked me what I was doing. I said loitering within tent.

Knock-knock!

Who’s there?

Ash.

Ash who?

Bless you!

A photo shared by a friend.

When I talk with fellow gardeners the word 'perennial' comes up over and over again.

This elegant little bird, that visited our garden today, is a young moorhen. It was unafraid of us and came really close to take food we put down. We're hoping it will return tomorrow.