## Listener No 4574, *Well-spoken*: A Setter’s Blog by Miles

Posted by Listen With Others on 20 Oct 2019

Some years ago I became aware of the 9-point circle, a property of any triangle except isosceles (8 points) or equilateral (only 6), and I thought it involved some really elegant geometry. More recently I wondered if it could be incorporated into a crossword and make a diverting change from the usual verbal gymnastics. After much exploration, I discovered the triangle with vertices at (0,0), (10,0) and (6,12) which had critical points with predominantly integer coordinates. This triggered the 13×11 grid shape. Using O for each point on the 9-point circle and X for each vertex of the triangle seemed a neat idea, and presented a nice challenge for the grid-fill.

However, the Euler line was, for me, just too irresistible to ignore, and for this to feature, extra constraints were imposed, and these pleasingly were not insurmountable. QIGONG was certainly involved at one stage, opposite perhaps NUTTER but other options prevailed. Likewise ELAN and STEP, ALA and TUE, LEO and EEN all had ‘their day’, but, for reasons that I still do not fully comprehend, maximising the average word-length and minimising the number of fully checked entries are paramount nowadays, so they had to be dumped.

As so often happens in a crossword construction serendipity intervened and I spotted the anagram of LONE TEPEE LEANS, which I thought was a pleasing bonus. This was the reason for using a carte blanche, requiring solvers to recognise the four unclued entries by filling in bars, but the editors warned me of the resulting ‘clutter’ interfering with the more significant drawing, so bars were not required and they were (probably) right.

The cluing, for me, usually takes longer than the grid construction. I recall that the clue for ANNEX originally had 22.5 degrees, but with the need for omitted letters, this rather conveniently was adapted to 45 degrees (without roasting alive any Floridans !). The clue with the Man U legend was dedicated to my brother who lives in the Manchester area. I particularly liked the clue ‘Spring: this describes about a quarter of years’ for LEAP, as an example of the same words capable of being interpreted in two distinct ways. I have to give credit to the editors for introducing the de Longchamps reference in the ETEN clue, as, until then, I was unaware of his connection with the Euler line.

As for the title, it was meant to hint at OILER, a well, but I was unsure whether the mathematician’s name sounded like that or more like YEW-LER. However, the editors were happy, checking in Collins and ODE apparently.

The need to draw the line precisely through the centre of the circle was I felt a fair requirement to confirm that solvers were fully aware of what was going on, though I can see that some non-mathematicians may beg to differ. However, mathematicians could check that the 12 points (3 X’s and 9 O’s) were all precisely positioned, centre of cell, as was the orthocentre, also on the Euler line at the G cell in row 11, confirmed by drawing the 3 altitudes through it from each X. Furthermore if (6,12) is called point A and (0,0) is B and (10,0) is C, then tan A = 1, tan B = 2, tan C = 3 and angle A = 45 degrees.

And finally (paying homage to Columbo), here is a challenge to acute observers. 50 is the smallest number expressible as the sum of two squares in two different ways. How is this exploited in the geometrical construction?

## Alan B said

Well, it’s 23 October now, so I’ll come on with my twopenn’orth.

First, I’ll just say what an amazing construction this was, with so many points of interest landing either exactly on the centre of a cell or exactly on a border between cells.

The points of interest are more than the three vertices of the scalene triangle, the nine points on the circle and the centre of the circle (marked with an X). There are also the circumcentre of the triangle and the orthocentre of the triangle. The Euler line is the line drawn through the circumcentre (the P of CATNEP in the grid) and the orthocentre (the G of NEARSIGHTED). (The centroid, or ‘centre’, of the triangle, incidentally, is one third of the way along the Euler line from the circumcentre to the orthocentre and does not have integer coordinates.) The centre of the circle (the X added at the end) is halfway between the circumcentre and the orthocentre.

The property of 50 that it is the sum of two squares in two different ways is exploited throughout this scenario. The circumcentre (P) is an x distance 5 and a y distance 5 from the two vertices of the triangle along the bottom and an x distance of 1 and a y distance of 7 from the vertex at the top. And when you look at the radii of the circle you will find that these linear distances are halved: 2.5/2.5 and 0.5/3.5.

Congratulations. Just a pity that Year 11/12 maths is needed to discover all these thematic shapes, points and lines, unless one can learn all about them in a hurry in time for the deadline!