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is said that projective geometry is a science born of art (Morris Kline, "The science born of art proved itself an art") and rightly so. The only problem is that the appreciation of art requires a long apprenticeship. The same is true of geometry. (Forget, even for a moment, the false promises of populism theoretical pedagogical fads in education - in its perennial quest for the philosopher's stone that would make all learning in a "blowout.")

The problem then be assumed known pose the following theorem:

If a triangle ABC are taken in BC points P, Q and R on AB CA so that the lines QR, RP and PQ intersect BC, CA, and AB in points P ', Q' and R 'respectively, then the points P', Q 'and R' are collinear if and only if the lines AP, BQ and CR are concurrent. (Exercise of it, prove it by applying the theorem of Menelaus.)

The problem I wish to raise here is:

If (under the conditions of the theorem above) the lines AP, BQ and CR are concurrent What can be said of lines AP, BQ 'and CR'?

Solution:

is easy to see that the answer is that they are concurrent. But this answer only comes after you get to see that besides the triangle PQR, there is another triangle that meets the PQR same conditions, except that is outside the triangle ABC. This triangle is the PQ'R '.

See

A -> P in BC .......................... A -> P

B-BC -> Q in CA .......................... B -> Q 'in CA

C -> A in AB .. ........................ C -> R 'in AB


AP, BQ and CR concur ......... ....... AP, BQ 'and CR' concur

......................... iff iff

'............................. AB.PQ AB.PQ = R '= R
BC.QR = P' ............................. BC.Q 'R' = P '= Q'.....
CA.RP ........................ CA.R 'P = Q

aligned ................ aligned ............

But R, Q and P 'are collinear by definition, since P' is intersection of BC and QR and therefore is on the line QR.

But what I want to emphasize here is that solving these problems (or prove these theorems) is a task that, plus it is very time consuming, requires knowledge of geometry have also needed to devote time for learning. And if the teen is believed the promise of populist educators will never be willing to spend more than 5 minutes to solve a math problem. From this point of view we can say that the school promotes ignorance.

Everybody Seems to Think I'm lazy
I do not mind, I think they're crazy Running everywhere at
Such a speed
Till They find, There's No Need

(I'm Only Sleeping, The Beatles)

Finally let me talk it I walked in Mexico City the week of the Congress of the Mexican Mathematical Society (24-28 October) and, after living three days in the abstract world in the Zocalo found a way to unwind and return to postmodernism: I chose the psychiatry of a sorcerer Aztec alternative.