Thales' Theorem, Perpendicular Line, Isosceles and Congruent Triangles

Thales' Theorem: Given a circle, any triangle involving the diameter and a non-collinear point on the circle, there will be a right triangle with the right angle being at the third point.

The top circle will illustrate this. B and C can be manipulated.

REFER TO THE BOTTOM CIRCLE
You can manipulate E and F.

Notice:
1) Line Segment EE' is a diamater of the circle.
2) Points F and G are on the circle. We see the same 90° angles.
3) Line Segment FG is perpendicular to the diameter.

Notice that Triangles EFE' and EGE' are congruent. This can be verified using Side-Angle-Side, Side-Side-Side, or Hypotenuse-Leg.

Notice that even when segment FG is a chord and not a diameter, Triangle EFG is an isosceles triangle with congruent sides EF and EG.

Further, if Point F is moved so that Segment FG goes through Point D, the two colored triangles
1) become congruent isosceles triangles, and
2) make a square.

Darron Steele, Created with GeoGebra