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
30 É. Sashalmi, M. Hoffmann Figure 3. Theorem 7. Consider the rectangle ABC D and one of its inner points P. Let the pedals of P on the sides AB , BC , CD and DA be Pi, P 2-, Ps and P4, respectively. If we build similar rectangles on the segments of the sides defined by the pedals in a way, that the larger sides of the rectangles are all parallel to the larger side of the original one, then the sum of the areas of the rectangles erected on the segments A Pi, B P2 , CP3 and DP.\ equals the sum of the rectangles erected on the segments P\B , P2C, P3D and P4A. Moreover, the rectangle bounded by the lines containing the outer sides of the left " rectangles is homothetic to the original one and the ratio of homothety is A = 2. Similar statement holds for the rights rectangles. Proof. The first part of the statement can be proved analogously to Theorem 3 and 6. For the ratio of homothety let us denote the ratio of the two sides of the rectangle by p = Consider the "left" rectangles. The sides of these rectangles parallel to AB are AP U pBP 2, CPs and pDP 4 (c.f. Fig. 4). The side A!B' of the large rectangle parallel to AB is the sum of these sides: A'B' = APi + pBP 2 + CPs + pDP 4, but APi + CP 3 = AB, while pBP 2 + pDP 4 = pBC = AB, thus A'B' = 2 AB. Similarly B'C' = 2BC and this was to be proved.
