angle in a semicircle

there's a neat proof of this circle theorem in Paul Lockart's article
('A Mathematician's Lament')









Paul's article is worth a read, or a re-read, arguing that maths is fundamentally about problems and these must be made the focus of a student's mathematical life, however painful and frustrating this may be

he describes a student's argument that if you take a right-angled triangle (hypotenuse passing through the centre) and rotate it around (through 180) it creates a four-sided shape, the sides of which must be parallel ...
so it is a parallelogram

further, the diagonals are both the same length ...
so it must be a rectangle

they complete their view that since the triangle got rotated halfway round the circle, one 'tip' ends up exactly opposite from where it started
that's why the other diagonal is also a diameter

and since it's a rectangle the angles are 90 degrees