This is a test whether Twitter can be useful to spot results. As a corollary of a Cauchy-Binet generalization for pseudo determinants, there is an identity for classical determinants. Its a special case of a special case but maybe this is known. Can twitter find out?
• Has anybody seen the identity det(1+A^2) = sum_P det^2(A_P) over all minors of a symmetric nxn matrix? (see http://goo.gl/5dQon ).

In the summer of 2013, some experiments with Twitter proofs on 21a twitter. These are proofs in less than 140 characters (some of which are done using Mathematica).
• The Lagrange identity |v|^2 |w|^2 - (v.w)^2 = |v x w|^2 which the last tweet has proven, also implies Cauchy-Schwarz! Why? You can see it.
• Proof of Lagrange identity (a special case of Cauchy-Binet): v={v1,v2,v3}; w={w1,w2,w3}; u=Cross[v,w]; Simplify[(v.v) (w.w) - (v.w)^2==u.u]
• The isosceles triangles in the 3:4:5 triangle have relative 1,2,3 ratios. Their height/base ratios is 1,2,3 (observation of Elijah Gunther).
• a={0,0};b={4,0};c={0,3}; x={1,0};y={8,9}/5;z={0,1}; {u,v,w}={x+y,y+z,z+x}/2; 2{Norm[a-w]/Norm[x-z],Norm[c-v]/Norm[y-z],Norm[b-u]/Norm[x-y]}
• By defining dot product, proving Cauchy-Schwarz, defining angle, we got Al Kashi and the cos formula. For right angles, we get Pythagoras.
• Can define alpha by cos(alpha) = v.w/(|v| |w|). Then (v-w).(v-w) = |v|^2 + |w|^2 - 2 v.w = |v|^2+|w|^2-2 |v| |w| cos(alpha) (Cos Formula)
• Define a=v.w. Assume |w|=1 without loss of generality. 0 <= (v-a w).(v-a w) = |v|^2-a^2 shows v.w = a <= |v| = |v| |w| (Cauchy-Schwarz)

It seems that Twitter is not very much used in higher education yet. I tried it out first in fall 2009 in math21a, then in spring 2010:

 Fall 2009. Local Copy [TXT]. Spring 2010 Local Copy [TXT] .