Geometric Construction 14
In this drawing:
show that the point Q is located at
.
Show that if
then h(b)=g(b),h(a)=f(a). If in addition we also have f(c)=g(c), then f(c)=h(c)=g(c).
(Aitken's Lemma)
Given Q_{1}(x_{1},y_{1}),Q_{2}(x_{2},y_{2})
and Q_{3}(x_{3},y_{3}),
construct the graph of the polynomial p(x) satisfying
p(x_{1}) = y_{1}, p(x_{2})
= y_{2}, p(x_{3}) = y_{3}.
Given (x_{1},y_{1}), (x_{2},y_{2}),
(x_{3},y_{3}) and (x_{4},y_{4}),
construct the graph of the polynomial p(x) satisfying
p(x_{1}) = y_{1}, p(x_{2})
= y_{2}, p(x_{3}) = y_{3},
p(x_{4}) = y_{4 }.
Given four points p_{0}, p_{1},p_{2},p_{3},
we are to construct the Bezier cubic curve passing through these
points as follows:

Construct p_{01 }= (1t)p_{0 }+ tp_{1}

Construct p_{12 }= (1t)p_{1 }+ tp_{2}

Construct p_{23 }= (1t)p_{2 }+ tp_{3}

Construct p_{012 }= (1t)p_{01 }+ tp_{12}

Construct p_{123 }= (1t)p_{12 }+ tp_{23}

Construct p_{0123 }= (1t)p_{012 }+ tp_{123}
The locus of p_{0123 }forms the required curve.
Show that
Construct the parabola tangent to two given straight lines at two given
points on each of the lines.