Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)
SASHALMI, É. and HOFFMANN, M., Generalizations of Bottema's theorem on pedal points
26 É. Sashalmi, M. Hoffmann Since for the Brocard angle a 2 + b 2 + c 2 COt UJ — — (1) holds (c.f. [4]), the ratio of the homothety in Theorem 2 can simply be written as A = 1 -j- cot u>. Throughout the paper we use the phrases "left" and "right" to distinguish the two families of squares or other builded polygons. 2. New results on triangles At first we prove that Bottema's statement holds not only for squares but for any rectangles similar for each other and also for regular triangles. Then we examine the ratio of homothety of Theorem 2 in the case when the squares are erected onto the inner side of the triangle and show that it equals cotu; — 1. Theorem 3. Consider the triangle ABC and one of its inner points P. Let the pedals of P on the sides AB, BC, CA be Pi , P 2 and P 3, respectively. If we build similar rectangles on the segments of the sides defined by the pedals, then the sum of the areas of the rectangles erected on the segments AP\, BP 2 and CP3 (i.e. the "left " rectangles) equals the sum of the rectangles erected on the segments P\B, P 2C and P 3A (i.e. the "right" rectangles). Proof. Here we use the basic idea of [3]. Let us denote the sides of the triangle by a, b,c and the segments defined by the pedals by the following: c/ = AP\; c r = PiB; ai = BP 2 ; a r — P 2C ; b[ = CP3; b r = P 3A. From Theorem 1 it is follows, that af + bf + cf = a 2 + b 2 r+c 2 r. (2) Let us denote the other side of the rectangle erected onto at by s and let p = Thus the area of this rectangle can be written as ais = aipai = a 2p. Since the rectangles are similar to each other, p is the ratio of their sides for all rectangles. Thus the sum of the areas of the "left" rectangles is afp + b 2p + cjp = p(af + bf + c 2). Similarly for the "right" rectangles alp -\-b 2,p-[- c 2,p = p(a 2. + b 2 + c 2) holds, which, together with (2) proves the statement.
