P-P Situation 6: Problem 12.
The soved problem:
In each semicircle of the arbelos are inscribed triangles with an acute angle of two of the vertices of which are at the end of the diameters of the respective semicircles. Prove that
where are the sides of the triangles , respectively.Solution:
The triangles set in this way are rectangular because two of their vertices are parts of the diameters of the semicircles. Therefore, they are similar. Hence:
Problem 12.
In each semicircle of the arbelos are inscribed triangles with equal angles, two of the vertices of which are at the end of the diameters of the respective semicircles. For the elements of the triangles prove that if
then
, , ,
where are the radii of the inscribed circles, – radii of the circumscribed circles, – semiperimeters, – heights to the hypotenuses, - areas of the triangles.