Try this Drag any orange dot. The angle ∠QRP will always be a right angle.

Put another way: If a triangle has, as one side, the diameter of a circle, and the third vertex of the triangle is any point on the circumference of the circle, then the triangle will always be a right triangle.

In the figure above, no matter how you move the points P,Q and R, the triangle PQR is always a right triangle, and the angle ∠PRQ is always a right angle.

The converse of Thales Theorem is useful when you are trying to find the center of a circle.
In the figure above, a
right angle
whose
vertex
is on the circle *always* "cuts off" a diameter of the circle. That is,
the points P and Q are always the ends of a
diameter line.

Since the diameter passes through the center, by drawing two such diameters the center is found at the point where the diameters
intersect.

For an animated demonstration of this technique see Find the Center of a Circle with a Right-angled Object.

- Circle definition
- Radius of a circle
- Diameter of a circle
- Circumference of a circle
- Parts of a circle (diagram)
- Semicircle definition
- Tangent
- Secant
- Chord
- Intersecting chords theorem
- Intersecting secant lengths theorem
- Intersecting secant angles theorem
- Area of a circle
- Concentric circles
- Annulus
- Area of an annulus
- Sector of a circle
- Area of a circle sector
- Segment of a circle
- Area of a circle segment (given central angle)
- Area of a circle segment (given segment height)

- Basic Equation of a Circle (Center at origin)
- General Equation of a Circle (Center anywhere)
- Parametric Equation of a Circle

- Arc
- Arc length
- Arc angle measure
- Adjacent arcs
- Major/minor arcs
- Intercepted Arc
- Sector of a circle
- Radius of an arc or segment, given height/width
- Sagitta - height of an arc or segment

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