By Kiran Kedlaya
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An alternate description of the projective plane turns out to be quite useful, and corresponds more closely to the artists’ conception. View the Euclidean plane as some plane in three-dimensional space, and fix a point O not on the plane (corresponding to the eye). Then each point on the plane corresponds to a line through O passing through that point, but not 46 all lines through O correspond to points on the Euclidean plane. In fact, they correspond to the points at infinity. In other words, we can identify the projective plane with the set of lines in space passing through a fixed point.
2. In triangle ABC, let G, I, N be the centroid, incenter, and Nagel point, respectively. Show that G, I, N lie on a line in that order, and that N G = 2 · IG. ) 3. Let P, Q, R be the midpoints of sides BC, CA, AB, respectively. Show that the incenter of P QR is the midpoint of IN . 5 Isogonal conjugates Two points P and Q inside triangle ABC are said to be isogonal conjugates if ∠P AB = ∠QAC and so on. In other words, Q is the reflection of P across each of the internal angle bisectors of ABC.
G. 4). A general principle behind this method is that problems with few circles are easier than those with many circles. Hence when inverting, one should find a “busy point,” one with many circles and lines going through it, and invert there. 3 1. Make up an inversion problem by reversing the paradigm: start with a result that you know, invert about some point, and see what you get. The tricky part is choosing things well enough so that the resulting problem doesn’t have an obvious busy point; such a problem would be too easy!