CTK Insights

27 Mar

Weekly report, March 21-27, 2016

This week a discussion on tweeter, brought to mind a quote by Underwood Dudley I used years ago

Can you recall why you fell in love with mathematics? It was not, I think, because of its usefulness in controlling inventories. Was it not because of the delight, the feeling of power and satisfaction it gave; the theorems that inspired awe, or jubilation, or amazement; the wonder and glory of what I think is the human race's supreme intellectual achievement? Mathematics is more important than jobs. It transcends them, it does not need them.

Is mathematics necessary? No. But it is sufficient.

With nine pages - solved problems and proved math statements - added to my site this week, I truly have a very good reason to mention that quote.

In the prior week, Galina Gavrilenko - a Russian mathematics teacher -has informed me of a theorem that she discovered, and that long searches on the web made her believe that the theorem was new.

Let similarly oriented squares ABGH, BCIJ, CDKL, and ADEF be erected on the sides of a parallelogram ABCD. Assume U,V,W,Z are the midpoints of segments FH, GJ, IL, and KE, respectively.

Squares on the sides of a parallelogram

Prove that the quadrilateral UVWZ is a square.

The theorem was obviously related to a few well-known and popular results in geometry: Finsler-Hadwiger theorem, van Aubel's theorem, Vecten's configuration, Thebault's first problem. With such a variety of possible connections, it seemed rather implausible that the statement has been overlooked so far. But my web searches also failed to find a precedent. With that, I set out to prove that theorem, found three simple but independent proofs, with confirmed connections to the better known problems. Finding the proofs was a delight in itself; it was augmented by the realization that, although the connections to the older problems have been really very strong and direct, Galina's discovery was in all likelihood indeed new.

Another enjoyable piece of mathematics has been supplied by Miguel Ochoa Sanchez from Peru.

AreaInequalityInSquare

He formulated a statement for a square, but his beautiful proof worked equally well for trapezoids. It was a pleasure to come up with this realization. Sometimes problem posers insert into their formulations unnecessary details that may obscure the real focus of the problem. These are commonly referred to as red herrings. Miguel is a great inventor of geometric problems; I am quite confident he chose a more specific configuration because it suggested applicability of analytic methods thus leading away from his wonderfully simple and general geometric argument.

I also received an attractive statement from Teo Lopez Puccio from Argentina - an aspiring student of mathematics. The statement is related to the configuration of arbelos, the shoemaker's knife, made famous yet by Archimedes.

ArbelosMorsels

In the two diagrams, the blue and yellow figures have the same areas, to check which is a simple exercise.

My Romanian correspondents continued the stream of various inequalities. Dorin Marghidanu offered an inequality involving the three altitudes of a triangle:

InequalityInTriangle

Solutions came from Japan (Kunihiko Chikaya); Greece (Vaggelis Stamatiadis); Romania (Dorin Marghidanu's and separately by Leo Giugiuc.) Leo Giugiuc was also a partner with Daniel Sitaru in coming up with another inequality

ApplicationOfSchursInequality2

The two solutions (one by Imad Zak from Lebanon) were both based on the famous and very useful Schur's inequality.

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