Can any right triangle be inscribed in a circle?

Question

Can every right triangle be inscribed in a circle?

If so, we could infer: ": in any right triangle, the segment which cuts the hypotenuse in two and joins the opposite angle is equal to half the hypotenuse". (as $r=\frac{1}{2d}$).

Thanks!

Answer 1

Yes. Take the hypotenuse as the diameter. Thales' theorem then tells you that the right-angle vertex lies on the circle itself.

Answer 2

You are mixing up two different, unrelated questions.

• Any triangle can be inscribed in a circle.

• Right triangles are the only ones where the circumcenter lies on one of the sides.

Answer 3

Given any three non-colinear points, there is always a unique circle -the so called circumcircle- whose circumference comes through them. The center of this circle -called the circumcenter- can be found as the point of intersection of the perpendicular bisectors of the line segments defined by these three points.

So, any triangle (not necessarily right-angled) can be inscribed in a circle. In case it is right-angled then the circumcenter -by the above construction- lies on the center of the hypotenouse.