promień okręgu Marta: Znajdź promień okręgu opisanego na czworokącie ABCD

Basia: sinα=²_2r=¹_r sinβ=⁵_2r

	1		r2−1
cos2α=1−		=
	r2		r2

	25		4r2−25
cos2β=1−		=
	4r2		4r2

α+β=60 cos(α+β)=cos60=¹₂

	√r2−1		√4r2−25		1		5
cos(α+β)=cosα*cosβ−sinα*sinβ=		*		−		*		=
	r		2r		r		2r

√r2−1*√4r2−25−1
2r2

√r2−1*√4r2−25−5		1
	=
2r2		2

√r²−1*√4r²−25−5=r² √r²−1*√4r²−25=r²+5 (r²−1)(4r²−25)=(r²+5)² 4r⁴−25r²−4r²+25=r⁴+10r²+25 3r⁴−39r²=0 3r²(r²−13)=0 r²−13=0 r²=13 r=√13

Basia: inny sposób: α=60 ⇒ β=120 z²=2²+5²−2*2*5*cos120 = 4+25−20*(−¹₂)=29+10=39 δ=2β=240 ⇒ γ=120 z² = r²+r²−2*r*r*cos120=2r²−2r²*(−¹₂) = 2r²+r²=3r² 3r²=39 r²=13 r=√13