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The Inscribed Angle Theorem and Thales

Stand on a circle and look across at a fixed arc: the angle you see is always the same, no matter where you stand — and it is exactly half the angle seen from the centre. That one fact, and its right-angle special case named for Thales, unlocks the whole circle.

Two ways to look at the same arc

In the previous guide you met the circle's furniture — its centre, radii, chords, arcs, and tangents. Now we put an angle inside it and watch something almost magical happen. Pick two points A and B on a circle; together they cut the circle into two arcs. There are two natural ways to 'see' the arc from A to B. From the centre O, the central angle angle AOB opens straight onto that arc, two radii fanning out to A and B. From a third point P sitting on the circle itself, the inscribed angle angle APB opens onto the same arc, but with its vertex on the rim rather than at the hub.

The central angle is the honest, plain measure of the arc: by definition, the arc's measure in degrees just is m(angle AOB). The inscribed angle is the interesting one, because P can slide all around the circle. Drag P along the major arc and you might expect angle APB to grow as you near the chord and shrink as you pull away — that is what intuition whispers. Intuition is wrong here, and being wrong in a clean, surprising way is exactly why this theorem is worth its name.

The theorem: the inscribed angle is half the central angle

Here is the statement in full. If an inscribed angle and a central angle subtend the same arc, then the inscribed angle is exactly half the central angle: m(angle APB) = (1/2) m(angle AOB). Two consequences fall out at once. First, since the central angle is fixed once A and B are fixed, the inscribed angle never changes as P wanders along its arc — every viewpoint on that arc sees the chord AB under the very same angle. Second, the number is always a clean half. An arc that the centre opens to 100 degrees is seen from the rim as 50 degrees, full stop.

Why is it half? The cleanest case is when one side of the inscribed angle, say PB, runs straight through the centre O — a diameter. Then look at triangle OPA. Both OP and OA are radii, so they are equal, making triangle OPA isosceles; by the isosceles triangle theorem the two base angles are equal, so m(angle OPA) = m(angle OAP). The central angle AOB is an exterior angle of this triangle, and the exterior-angle theorem from the Triangles rung says it equals the sum of the two remote interior angles — which here are equal — so m(angle AOB) = 2 m(angle OPA) = 2 m(angle APB). Halve it and you have the result.

Thales: the half-circle right angle

Now feed the theorem its most famous special case. Let AB be a diameter rather than an ordinary chord. The central angle for a diameter is a straight angle: A, O, B lie in a line, so m(angle AOB) = 180 degrees. Any point P on the circle sees this diameter under an inscribed angle of half that — m(angle APB) = 90 degrees. So every point on the circle (other than A and B themselves) looks back at the diameter at a perfect right angle. This is Thales' theorem: an angle inscribed in a semicircle is always a right angle.

A ----------- O ----------- B      AB is a diameter, O the centre
  \                       /
   \         P           /        P is any point on the circle
    \     (on circle)   /
     \                 /          angle AOB = 180 deg  (A,O,B in a line)
      \               /           angle APB = 90 deg   (half of 180)
       \____________/             so triangle APB is right-angled at P
Thales: the angle subtended by a diameter from the rim is always 90 degrees.

This is a genuine two-way street, and the converse is just as useful. If a triangle ABP has a right angle at P, then P must lie on the circle whose diameter is the hypotenuse AB — its centre is the midpoint of AB, and OP, OA, OB are all equal radii. So the set of all points that see a fixed segment at a right angle is precisely a circle on that segment as diameter. That converse is the workhorse: it turns 'there is a right angle here' into 'these four points lie on one circle', the bridge into the next guide on cyclic quadrilaterals.

Putting the theorem to work

Let us walk a small numerical example all the way through, because the bookkeeping is where the theorem earns its keep. Suppose A, B, C, P all sit on one circle with centre O, and the central angle on arc AB is m(angle AOB) = 70 degrees, while the central angle on arc BC is m(angle BOC) = 90 degrees. We want the inscribed angle m(angle APC), where P sits on the major arc so that arcs AB and BC together face it.

  1. The arc from A to C (through B) has central angle m(angle AOC) = 70 + 90 = 160 degrees, since the two central angles are adjacent and add.
  2. P sees this same arc AC, so its inscribed angle is half: m(angle APC) = (1/2)(160) = 80 degrees.
  3. Check a piece: an inscribed angle on arc BC alone would be (1/2)(90) = 45 degrees, and on arc AB alone (1/2)(70) = 35 degrees; 45 + 35 = 80, consistent.

Two refinements make this even more powerful in practice. A tangent can stand in for one arm: if instead of a chord you let a tangent line touch the circle at A and measure the angle it makes with chord AB, you get the tangent-chord angle, which equals half the intercepted arc just like an inscribed angle — think of it as the limiting case where P slides all the way to A. And when P sits on the opposite arc, it sees the chord under the supplement, half of the reflex central angle; that complementary fact is precisely what makes cyclic quadrilaterals have opposite angles summing to 180 degrees.

Why it matters, and where it leads

Step back and savour what you have. The inscribed angle theorem says a circle is a place of perfect agreement: every observer standing on a given arc measures the chord across from them at one shared angle. That is why a circle, and only a circle, is the locus of points seeing a fixed segment at a fixed angle — a fact that quietly governs everything from camera framing to the geometry of the goal in football. The right-angle special case, Thales, is the seed of coordinate geometry's circle facts and of countless construction tricks.

The next guide cashes in the two refinements just hinted at. Build a quadrilateral whose four corners all lie on one circle and the inscribed angle theorem forces its opposite angles to be supplementary — the defining property of a cyclic quadrilateral. Push a little further and the same circle-and-angle reasoning produces the power of a point, a single number that ties together intersecting chords, secants, and tangents. Everything in that final guide is this one theorem, viewed from new angles — quite literally.