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

28 É. Sashalmi, M. Hoffmann which, together with the previous equation yields Oj_ _ bj_ _ Q _ cot / ~ g - h~

C° Thus the ratio of homothety is ai - f bi - g ci-h A = - = = — COT U) — I, / 9 which completes the proof. By applying this method one can prove several similar theorems and compute the ratios of homothety. Here we mention only one more example (see Fig. 2). Figure 2. Theorem 5. Consider the triangle ABC and one of its inner points P. Let the pedals of P on the sides AB, BC,CA be P\, P2 and P3, respectively. If we build regular triangles on the segments of the sides defined by the pedals, then the sum of the areas of the triangles erected on the segments AP\,BP2 and CP3 equals the sum of the triangles erected 011 the segments P\B, P2C and P3A

. Moreover, if we consider those vertices of the "left" triangles which are not on the sides of ABC and draw parallel lines to the sides of the original triangle through of them, then the triangle bounded by these lines is homothetic to ABC and the ratio of homothety is A = 1 + cote«;. Similar homothety holds for the triangle constructed from the "right" builded triangles.