Pólya on Mathematical Abilities.

Pólya interviewed by Jeremy Kilpatrick.

JK: How did you identify the students you had who were best in mathematics? You taught some students who were good in mathematics. How could you tell who were the best ones?

GP: Who was the best one, I can’t tell you.

JK: Well, among the best, how could you identify their talent? They were quicker?

GP: Anyhow, they asked good questions. So they found out something by themselves. And so on. There is no simple way—. You see, people are too different. Mathematicians are too different. There is no simple way of describing it. I don’t think so.

JK: What about people who are creative in mathematics as opposed to just being able to learn it? What does that take? What does that require? Just great interest?

GP: I don’t know.

JK: Not everyone could be creative in mathematics.

GP: I said somewhere, “What is the difference between productive and creative?” If you think about a problem, if you produce a result, then you are productive. If in working you get into a method with which you can solve also other problems, then you are creative. That’s the difference. And that is difficult to say. I don’t think there are obvious signs to recognize this. I don’t think so.

Pólya on Mathematical Abilities. Jeremy Kilpatrick.

Archimedean Siblings out of Wedlock, i.e., Arbelos

Description and construction

Circle and Lucas Cubic

This is a generalization of proposition 2 in my paper  A Note on Reflections.

On June, 10, 2014, I posed a problem at ADGEOM, which was generalized by Angel Montesdeoca in the following form:

Let ABC be a triangle. Let P be an arbitrary point on the plane of ABC. Reflect P around the vertices of cevian triangle X_aX_bX_c of a point X. This give us the triangle X'_aX'_bX'_c. Reflect the triangle

X'_aX'_bX'_c around sides of triangle ABC. Let [X_P] be the circumcircle of the triangle so formed.

P lies on the circle [X_P] if and only if X lies on the Lucas cubic.

Then, X_aX_bX_c is the pedal triangle of a point Y (on Darboux cubic) and Y is the center of [X_P].

Proof (in Spanish)

Concurrent Circles

Let ABC be a triangle.
Let P be a point on the plane of ABC.
Let a, b, c, are the sides opposite vertices A, B, C.
Let P_a be the reflection of P around the perpendicular
bisector of a. Define P_b, P_c cyclically.

Then, circles (AP_bP_c), (BP_aP_c), (CP_aP_b) are concurrent at
the circumcircle of ABC.

Proof

A Note on Reflections

Click here to read the article. 

Circumbicevian Perspector

Here's another point associated to cevians. I have dubbed it "Circumbicevian" perspector.

Description and proof

Novel Concurrency of Perpendicular Bisectors

I dedicate this problem to my friend Anastasia Saychuk.

Description and proof