The Angle in a Semicircle

Take any circle and draw a diameter — a straight line right through the centre, with both ends on the circle. Pick any other point on the circle and join it to the two ends of the diameter. The angle you make there is always a right angle:

\angle APB = 90^\circ

It is the angle at the centre theorem in a special case: the diameter is a straight line, so the angle at the centre is 180^\circ, and the angle on the circle is half of that.

If AB is a diameter of a circle:

for any other point P on the circle, the angle \angle APB = 90^\circ;
it is the angle at the centre theorem in the special case where the centre angle is a straight line (180^\circ), so the circumference angle is half — 90^\circ;
so any triangle drawn on a diameter is right-angled.

Why it works

No measuring needed — the diameter does all the work. Step through the reason.

Wherever you slide P around the circle, the diameter keeps the centre angle at a straight 180^\circ, so the angle at P stays at 90^\circ.