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

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) £5 31 GENERALIZATIONS OF BOTTEMA'S THEOREM ON PEDAL POINTS Eva Sashalmi and Miklós Hoffmann (Eger, Hungary) Abstract. Given a polygon and one of its inner points P, the orthogonal projections of P onto the sides of the polygon are called pedal points of P. Here we prove different results concerning configurations by attaching different types of polygons to the segments of the sides defined by the pedals. These theorems can be considered as the generalizations of Bottema's classical theorem. 1. Introduction Consider a triangle ABC and one of its inner points P. Let the orthogonal projection of P onto the sides AB, BC,CA be Pi, P-2 and P3, respectively. These are the pedal points of P. If we build squares on the segments of the sides defined by the pedals (outside of the triangle), we obtain six different squares. In [1] Bottema proved the following theorem about the areas of these squares: Theorem 1. The sum of the areas of the squares erected on the segments A P\

, BP2 and CP3 equals the sum of the squares erected on the segments P\B, PzC and P3A. More recently van Lamoen and other studied similar configurations ([2], [3]) and showed the following in [3]: Theorem 2. Let A\B\C\ be the triangle bounded by the lines containing the sides of the squares opposite to AP\

, BP2 and CPs. Similarly let, A2B2C2 be the triangle bounded by the lines containing the sides of the squares opposite to P\ B. P

2C and P3A. These two triangles are each homothetic to ABC and the ratio of homothety is a

2 + b

2 + c

2 41 ' where a,b,c are the sides and t is the area of ABC. To simplify the equation we use the following notations: Definition. The Brocard point ft and the Brocard angle w of ABC is t he point and angle for which l ABÜ = Z BCQ = ICAQ = to.