[justify][color=#666666][i]In each semicircle of the arbelos are inscribed triangles [math]\Delta,\Delta_1,\Delta_2[/math] with an acute angle of [math]30^\circ[/math][/i][i] two of the vertices of which are at the end of the diameters of the respective semicircles[/i][/color][color=#666666][i].[br]Prove that[/i][br][/color][center][math]a:a_1:a_2=b:b_1:b_2=c:c_1:c_2=n,[/math][/center][color=#666666]where [math]a,b,c; a_1,b_1,c_1;a_2,b_2,c_2[/math] are the sides of the triangles [math]\Delta,\Delta_1,\Delta_2[/math], respectively. [/color][/justify]

[justify][i][color=#666666]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: [br][/color][/i][/justify][center][math]a:a_1:a_2=b:b_1:b_2=c:c_1:c_2.[/math][/center]

[justify][i][color=#666666]In each semicircle of the arbelos are inscribed triangles [math]\Delta,\Delta_1,\Delta_2[/math] 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 [math]\Delta,\Delta_1,\Delta_2[/math] prove that if [br][/color][/i][/justify][center][i][math]a:a_1;a_2=b:b_1:b_2=c:c_1:c_2=n,[/math][/i][/center][i][color=#666666]then[br][/color][/i][center][i][color=#666666][math]r:r_1:r_2=n,[/math][br][br][math]R:R_1:R_2=n[/math],[br][br][math]s:s_1:s_2=n[/math],[br][br][math]h_{\left(c\right)}:h_{\left(c_1\right)}:h_{\left(c_2\right)}=n[/math],[br][br][math]F:F_1:F_2=n^2,[/math][/color][/i][/center][i][color=#666666]where [math]r,r_1,r_2[/math] are the radii of the inscribed circles, [br][math]R,R_1,R_2[/math] – radii of the circumscribed circles,[br][math]s,s_1,s_2[/math] – semiperimeters, [br][math]h_{\left(c\right)},h_{\left(c_1\right)},h_{\left(c_2\right)}[/math] – heights to the hypotenuses,[br][math]F,F_1,F_2[/math] - areas of the triangles.[/color][/i]