Sestrojte\ rovnoběžník\ ABCD,\ jsou-li\ dány\ jeho\ vrcholy\ A,B\
a\ zbývající\ vrcholy\ C,D\ leží\ na\ dané\ kružnici\ k(S;r).

Zadání

Diskuze o počtu řešení

Řešení

1) posun k (S, r) o vektor AB

Posunutím\ o\ \vec{AB}\ přejde\ bod\ D\ do\ bodu\ C\ resp.\ C\,'.\

2) průsečík k s k' = C resp. C'

3) posun C, C' o vektor BA

Zpětným\ krokem\ získáme\ bod\ D\ resp.\ D'.

Rovnoběžník ABCD resp. ABC'D'

S

k

A=D

B = C

k'

S'

S

k

A

B

k'

S'

S

k

A

B

k'

C

D

S

k

A

B

k'

C'

C

S'

S'

D

D'

a)\ 0\ řešení:\
I) |AB| = 2r ∧A,B∈k\

II)\ |AB| >2r

b)\ 1\ řešení:\
I)\ |AB| = 2r∧A,B∉k

c)\ 2\ řešení:\
I)\ |AB| < 2r\ ∧ A,B∉k

S

k

A = D'

B = C'

k'

S'

C

D

II)\ |AB| <2r\ ∧ A,B ∈k

S = (6.78, 0.2)
k = Circle(S, 2.5)
A = (6.78, -3.06)
B = (8.96, -2.5)
k' = Translate(k, Vector[A, B])
C_1 = Point(k')
D_1 = Point(k')
E = Midpoint(C_1, D_1)
F = Point(k')
G = Point(k')
H = Midpoint(F, G)
C', C = Intersect(k, k')
l = Segment(A, B)
m = Segment(B, C)
n = Segment(B, C')
D = Translate(C, Vector[B, A])
D' = Translate(C', Vector[B, A])
p = Segment(C, D)
q = Segment(D, A)
r = Segment(C', D')
s = Segment(D', A)
a = true
b = false
c = false
d = false
e = false
o = false
f = false
t = Radius(k)
text5: "r=" + (LaTeX[t]) + ""
v = Distance(A, B)
text6: "|AB| = " + (LaTeX[v]) + ""
I = (19.6, 5.48)
c_1 = Circle(I, 0.8)
J = Point(c_1)
f_1 = Line(J, I)
K = Intersect(c_1, f_1, 2)
d_1 = Translate(c_1, Vector[J, K])
L = Intersect(d_1, f_1, 2)
M = Midpoint(K, L)
N = (18.72, 3.68)
e_1 = Circle(N, 0.8)
g_1 = Line(N, xAxis)
O = Point(g_1)
O' = Mirror(O, N)
h_1 = Translate(e_1, Vector[O, O'])
P = Center(h_1)
S' = Center(k')
Q = (19.52, 1.3)
g = Circle(Q, 0.8)
R = (18.82, -0.1)
T = Point(Circle[R, 1.6])
h = Segment(R, T)
g' = Translate(g, Vector[R, T])
U = Intersect(g, g', 1)
U' = Translate(U, Vector[T, R])
V = (19.9, -3.02)
k_1 = Circle(V, 0.8)
W = (19.32, -4.4)
Z = Point(Circle[W, 1])
i = Segment(W, Z)
p_1 = Translate(k_1, Vector[W, Z])
A_1 = Intersect(k_1, p_1, 1)
B_1 = Intersect(k_1, p_1, 2)
E_1 = Center(g')
F_1 = Center(p_1)
G_1 = Translate(B_1, Vector[Z, W])
H_1 = Translate(A_1, Vector[Z, W])
j = Segment(Z, B_1)
i_1 = Segment(B_1, G_1)
j_1 = Segment(G_1, W)
l_1 = Segment(Z, A_1)
m_1 = Segment(A_1, H_1)
n_1 = Segment(H_1, W)
q_1 = Segment(T, U)
r_1 = Segment(U, U')
s_1 = Segment(U', R)
N_1 = (19.7, -1.12)
t_1 = Circle(N_1, 0.8)
I_1 = Point(t_1)
J_1 = Point(t_1)
c_2 = Translate(t_1, Vector[I_1, J_1])
K_1 = Center(c_2)
L_1 = Intersect(t_1, c_2, 1)
a_1 = Segment(I_1, J_1)
b_1 = Segment(L_1, J_1)
po: (p * o)
M_1 = Translate(L_1, Vector[J_1, I_1])
f_2 = Segment(L_1, M_1)
g_2 = Segment(M_1, I_1)