Geo-gebra

Geometry Level 3

Three unit circles are inscribed inside a rectangle as shown on the figure. The diagonal of the rectangle cuts three chords from the three circles of lengths Y,X,and Z respectively. Find the value of 6X² - 10YZ.


The answer is 0.

Aziz Alasha by Aziz Alasha , 59, Sudan

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2 solutions

Fletcher Mattox
Jul 15, 2020 
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    from sympy.geometry import Point, Line, Circle, intersection
    p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 5)

    p1 = Point(0,0)
    p2 = Point(6,2)
    diagonal = Line(p1, p2)

    c1 = Circle(Point(1, 1), 1)
    c2 = Circle(Point(3, 1), 1)
    c3 = Circle(Point(5, 1), 1)

    i1 = intersection(diagonal, c1)
    i2 = intersection(diagonal, c2)
    i3 = intersection(diagonal, c3)

    y = i1[0].distance(i1[1])
    x = i2[0].distance(i2[1])
    z = i3[0].distance(i3[1])

    print("x=", x)
    print("y=", y)
    print("z=", z)
    soln = 6*x*x - 10*y*z
    print("6x^2 - 10yz =", soln)


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x= 2
y= 2*sqrt(15)/5
z= 2*sqrt(15)/5
6x^2 - 10yz = 0

0 Helpful 0 Interesting 1 Brilliant 0 Confused
Aziz Alasha
Jul 19, 2017

Let the rectangle be ABCD , and the UNIT circles be C1,C2 and C3.
Let the coordinates of A be (0,0) , the Origin,
So far, Using co-ordinate geometry :
The equation of the circles C1 , C2 and C3 are :
(x-1)² + (y-1)² =1,
(x-3)² + (y-1)² =1,
(x-5)² + (y-1)² =1 , respectively,
The equation of the diagonal AC is :
Y = x/3,
Now we can get the points of intersection of each circle with the straight line AC, from which we get the cord lengths : y = √2.4 ,
x = 2 ,
Z = √2.4 ,
Then : 6X² - 10YZ =24 - 24 = 0 .




0 Helpful 0 Interesting 0 Brilliant 1 Confused

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