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

Generalizations of Bot tenia's theorem on pedal points 29 3. New results on polygons In this section we generalize Theorem 1 for convex polygons and prove some further results about quadrilaterals. Theorem 6. Consider the convex polygon A\A2..-A

n and one of its inner points P. Let the pedals of P on the sides A\A2,A

2A

3,...,A n — 1 A

n A j be Pi

, P

2,

..., Pn-i, P

n, respectively. If we build "left" squares onto the segments Ai Pi

, (i = 1,..., n) and "right" squares onto the segments /Vt<+ i

, (i = 1,

..., n — 1) and P

nAi, then the sum of the areas of "left" squares equals the sum of the area of "right" squares. Proof. Applying the phytagorean theorem for the triangles PAjPi one can write AiPi

2 = PAi

2 - PPi\ i=l,...,n. PiA

i+

l

2 = PA

i+

l

2 - PPi

2, i= l,...,n- 1 PnAi

2 = PA,

2 - PP

2. n n I-I i=l n— 1 = - PP 2) + PAi

2 - PP

n

2 i= 1 n- 1 = Y,

PiAi+l

2+PnA

1

2, i = 1 which completes the proof. The statement remains valid if the builded quadrilaterals are not squares but rectangles similar to each other as it was in the triangle case (c.f. the proof of Theorem 3). The statement of Theorem 6 can be seen for pentagons in Fig. 3. We have to remark, that if we consider the pentagons bounded by t he lines containing the sides of the squares parallel to the sides of the original pentagon, the two pentagons are not homothetic to each other. Generally speaking this property is valid only for triangles. For special cases, however, homothety still holds for quadrilaterals, as we will see in the next theorems. Similarly This yields