33. Construct a Triangle Given Its Base, Altitude to the Base and Product of the Other Two Sides

​
c
4
v
1
k
2.2
plot range
5
steps
1
2
3
show Cassini oval
This Demonstration constructs a triangle
ABC
given the length
c
of the base
AB
, the length
v
of the altitude from
C
to
AB
and the product
ab=
2
k
of the other two sides. Since the radius of the circumcircle
abc/2cv=
2
k
/2v
is a rational function of
k
and
v
, we can construct it using similar triangles.
Construction
1. On a horizontal line, draw points
A
and
B
so that
AB=c
and let
I
be in the interval
OB
such that
BI=1
. Let
O
be the midpoint of
AB
. On the vertical line through
O
, draw the point
N
so that
ON=v
. On the opposite side of the line
ON
, draw the points
M
and
R
so that
OM=k
and
OR=k/2
. Construct a point
P
on the horizontal line so that
PM
is parallel to
NI
. Then
OP/1=OM/ON=k/v
. Draw a point
S
on
ON
so that
PS
is parallel to
RI
. Then
2|SO/k=PO/1=
2
k
/2v
.
2. Now construct the center
T
of the circumcircle. There are two possibilities, one on either side of a horizontal line.
3. The point
C
is the intersection of the dashed line through
N
parallel to the horizontal line with either of the two circles. Since
cv=absinγ
,
sinγ=cv/
2
k
≤1
,
cv≤
2
k
.

Details

A Cassini oval (or Cassini ellipse) is a quartic curve such that if
P
is on the curve, the product of its distances from two fixed points
A
and
B
at a distance
c
apart is a constant
2
k
. Thus the original problem is equivalent to finding the intersections of the oval with a line parallel to the
x
axis at distance
v
.
Keeping
c
and
k
fixed and changing
v
, we get another construction of points on the Cassini oval.

External Links

Cassini Ovals (Wolfram MathWorld)
1. Constructing a Point on a Cassini Oval
2. Constructing a Point on a Cassini Oval
3. Constructing a Point on a CassiniOval
4. Constructing a Point on a Cassini Oval
1. Normal and Tangent to a Cassini Oval
2. Normal and Tangent to a Cassini Oval
32b. Construct a Triangle ABC Given the Length of AB, the Ratio of the Other Two Sides and a Line through C

Permanent Citation

Izidor Hafner, Marko Razpet
​
​"33. Construct a Triangle Given Its Base, Altitude to the Base and Product of the Other Two Sides"​
​http://demonstrations.wolfram.com/33ConstructATriangleGivenItsBaseAltitudeToTheBaseAndProductO/​
​Wolfram Demonstrations Project​
​Published: November 7, 2018