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3 37 A second example: are there points for which the distances are proportional to the opposite sides? The answer is that there are two such real points Q1 and Q2 only if a2 , b2 , and c2 satisfy the triangle inequalities, that is, if the triangle is acute. A completely diﬀerent application is the following. The condition under which two circles with radii R1 and R2 and distance d between their centers meet perpendicularly, is that R12 + R22 = d2 . If R2 = 0, the relation becomes d2 = R12 . That is, the center of a point circle which meets a true circle perpendicularly lies on that circle’s circumference.

4) if and only if ap, bq, and cr satisfy the triangle inequalities. 3 By what we have just seen, the distances r1 , r2 , and r3 from a point P to the vertices of a triangle ABC are unsuitable for use as a coordinate system. Their ratios cannot be used either: they correspond to two, one, or zero real points. They are no true rival for the trilinear or barycentric coordinates. However, the elaborate computations of this chapter have not been completely in vain. There are questions of geometric type for which they can be used.

6 have always attracted much attention from geometers. Closely related to them are considerations such as the following, which were stated by Minkowski (1903) [Min] for ﬁgures of very general type, and which we will try to present for the elementary case of a polygon. Let V1 = A1 A2 . . An and V2 = B1 B2 . . Bn be two n-gons whose corresponding sides are parallel: A1 A2 // B1 B2 , and so on. For simplicity, we’ve assumed in the ﬁgure that V1 lies inside V2 . This fact is not necessary for q3 P A3 p1 A1 B1 B3 A2 B2 Fig.