I encountered this problem in the "Elementary Geometry for College Students" book.
Diagram of a trapezoid (according to the book)
Some definitions from the book:
- When all vertices of a quadrilateral lie on a circle, the quadrilateral is a cyclic quadrilateral
- A trapezoid is a quadrilateral with exactly two parallel sides
- If the two legs of a trapezoid are congruent, the trapezoid is known as an isosceles trapezoid
- An isosceles trapezoid is cyclic
The book describes the process as such: "the center of the circle containing all four vertices of the trapezoid is the point of intersection of the perpendicular bisectors of any two consecutive sides (or of the two legs)."
I simply do not understand their instructions, I tried drawing a couple of diagrams as well as searching the net, but no luck. Sometimes the centre is found outside the trapezoid, which makes things even more confusing.
Could you help me understand the steps from their description (an image would be much appreciated) or maybe there is an easier algorithm, I would be interested in that too.
Best Answer
Since the trapezoid is inscribed into a circle, each side is a chord in that circle. The center of a circle does lie on a perpendicular bisector to any chord (because the center is equidistant from the ends of the chord). So you need to take two chords, draw perpendicular bisectors to them and the point of intersection of the bisectors will be the center. The center can be inside or outside of the trapezoid. You cannot use both bases as the perpendicular bisectors will coincide (i.e. you will not get a point of intersection).