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Seeking an integer perimeter for a particular cyclic quadrilateral, given its area and the radius of its circumcircle .

Step-by-step mathematics solution: Seeking an integer perimeter for a particular cyclic quadrilateral, given its area and the radius of its circumcircle .

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1. What the problem is asking

We have a convex cyclic quadrilateral (its four vertices lie on a circle) that

  • has two opposite right angles,
  • has area (2026), and
  • is inscribed in a circle of radius (R=40).

Among all quadrilaterals that satisfy those three conditions we have to decide

whether any of them can have an integer perimeter.

2. Geometry of a quadrilateral with opposite right angles

For a cyclic quadrilateral the opposite angles always add to (180^{\circ}).
Hence if one angle is a right angle, the opposite one must also be a right angle – the situation described in the problem.

Let the two right‑angled vertices be (B) and (D).
Because (\angle B) subtends the side (AC), the chord (AC) must be a diameter of the circum‑circle.
Similarly (\angle D) subtends the side (BD), so (BD) is also a diameter.

Thus the quadrilateral consists of the four endpoints of two diameters of the same circle.
The two diameters intersect at the centre (O) of the circle.

          B •
            \
             \      (diameter BD)
              \
               O •
              / \
   (diameter AC)   \
            /       \
          A •-------• C
                D •

Consequently both diagonals have the same length

[ |AC|=|BD|=2R=80 . ]

Let the angle between the two diameters be (\theta\;(0<\theta<\pi)).
Then the area of the quadrilateral is half the product of the diagonals multiplied by the sine of the angle between them:

[ \boxed{\ \text{Area}= \frac12\,(|AC|)(|BD|)\sin\theta =2R^{2}\sin\theta\ }. ]

3. Determining (\theta) from the given area

The area is (2026) and (R=40). Hence

[ 2026 = 2\cdot 40^{2}\sin\theta = 3200\sin\theta \quad\Longrightarrow\quad \sin\theta = \frac{2026}{3200}= \frac{1013}{1600}. \tag{1} ]

So (\sin\theta) is a rational number (but not an integer).

4. The side lengths in terms of (\theta)

Place the centre at the origin and the first diameter on the (x)-axis:

[ A=(-R,0),\qquad C=(R,0). ]

Let the second diameter make the angle (\theta) with the (x)-axis, i.e.

[ B=(R\cos\theta,\;R\sin\theta),\qquad D=(-R\cos\theta,\;-R\sin\theta). ]

The four sides are then

[ \begin{aligned} AB &= \sqrt{(R\cos\theta+R)^2+(R\sin\theta)^2} = R\sqrt{2+2\cos\theta}=2R\cos\frac{\theta}{2},\[2mm] BC &= \sqrt{(R-R\cos\theta)^2+(R\sin\theta)^2} = R\

Original question: Seeking an integer perimeter for a particular cyclic quadrilateral, given its area and the radius of its circumcircle . on Mathematics Stack Exchange, licensed CC BY-SA.

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