Thomas Downing [USA] (1928-1985) ~
‘Lambic Time’, 1962.
Acrylic on canvas (193 x 201 cm).uNiQuE ART HuNT wEEk ( XLIV ) :
Thomas Downing —~
“Thomas Downing received his BA in 1948 from Randolph-Macon College in Ashland, Virginia, and then moved to New York City to study at the Pratt Institute for two years. A grant from the Virginia Museum of Fine Arts provided him an opportunity to study in Europe at the Académie Julian in Paris.
Downing moved to Washington, DC, in the 1950s and studied under Kenneth Noland, one of the founding members of the Washington Color School, an association of the city’s Color Field painters.
Beginning in 1965, Downing taught at the Corcoran College of Art and Design for three years, where his ideas helped influence the next generation of Color School painters.”
~
ARTwoRks @ 9h AM CET
DAiLy — siNcE Oct 2015![ get to know the artist ★ buy me a coffee ]
The Fourth Side
Sweet triangle problem! Originally from B. F. Sherman’s The Fourth Side of a Triangle.
What information do we need to know to determine a triangle is as old as geometry questions get. If we know the incircle, circumcircle and 9 point circle, is the triangle determined? Vsauce made a YouTube short which is where I saw the problem.
A. P. Guinand showed that you cannot make a Euclidean construction of a triangle from that information.
All the sides of a triangle satisfy:
a) Each side has its endpoints on the circumcircle;
b) each side has its midpoint on the nine-point circle; and
c) each side touches the incircle.
It turns out there is a fourth segment that satisfies these conditions!
Sherman used some sweet calculations to come up with two methods to find this fourth side. Method 1: For a point M on the nine point circle, make the chord perpendicular to the orthocenter through M to the circumcircle. Find the point T on that line perpendicular to the incenter I. M, T, and O are 3 corners of a rectangle. The fourth corner P, when it is the inradius distance away from M (on the dotted red circle in this applet) indicates that the segment meets the conditions. Method 2: For a point U on the circle with diameter spanning the incenter and the orthocenter, construct a segment through the orthocenter, and mark points on that segment the inradius away from U. When either of those points is on the 9 point circle, that’s a segment that satisfies the conditions. This will happen 4 times. The three sides, and the fourth side!
Play in GeoGebra!
I enjoy this every time it comes up. I get how @galois-groupie had a dog in the fight (expression marked as ‘got to go’) but I think it would be a tough call for mathematicians.
- Ramanujan, here’s a blanket and free meals at a good Indian restaurant.
- Galois, not sure the glock helps, then he’s in a Napolean prison. Maybe relationship counseling?
- Abel, antibiotics for TB.
- Riemann, also TB.
- Ramsey, treatment for jaundice. (Probably better diagnoisis.)
- Clifford, also TB.
Y.Z. Kami, Endless Prayers IX, (mixed media on paper), 2015. From: Y.Z. Kami: Endless Prayers, LACMA – Los Angeles County Museum of Art, Los Angeles, CA, November 19, 2016 – March 19, 2017 [Gagosian, New York, NY. © Y.Z. Kami]
![arthuntblog:
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“ Thomas Downing [USA] (1928-1985) ~
‘Lambic Time’, 1962.
Acrylic on canvas (193 x 201 cm).
”
uNiQuE ART HuNT wEEk ( XLIV ) :
Thomas Downing —
~
“Thomas Downing received his BA in 1948 from Randolph-Macon College in...](https://64.media.tumblr.com/06b0dabdb952a842557a2b145277851f/tumblr_pwdl79KCwR1uhkf79o1_500.jpg)









