Converse of semicircle theorem?

[Delta]

How do you prove the converse of this theorem?

If a right triangle is inscribed in a semicircle then the hypoteneuse is the
diameter of the circle

[Lynn Kurtz]

Post by Delta
How do you prove the converse of this theorem?
If a right triangle is inscribed in a semicircle then the hypoteneuse is the
diameter of the circle

Converse: (?) If a triangle is inscribed in a circle with one side as
a diameter, then the triangle is a right triangle.

Proof is almost trivial. Let A and B be the ends of the diameter and C
the third vertex, with corresponding angles a, b, and c. Draw the
radius to C, forming two isosceles subtriangles. Then c = a + b and a
+ b + c = 180 so 2c = 180 and c is a right angle.

--Lynn

[Delta]

That is the theorem...but the converse is:

If a triangle is inscribed in a semicircle with the angle opposite to the
hypoteneuse as 90° then the hypoteneuse is the diameter...

How would you prove that ? I tried it many times and I get stuck

[matt grime]

Post by Delta
If a triangle is inscribed in a semicircle with the angle opposite to the
hypoteneuse as 90° then the hypoteneuse is the diameter...
How would you prove that ? I tried it many times and I get stuck

How about:

If the hypoteneuse isn't the diameter, then there is a diameter parallel
to it. consider the triangle formed by that diameter and the point on the
circle at the 90 degree angle. It's easy to see that it must be in fact
the original triangle else its angle there isn't 90. Which it must be.

[Lynn Kurtz]

Post by Delta
If a triangle is inscribed in a semicircle with the angle opposite to the
hypoteneuse as 90° then the hypoteneuse is the diameter...
How would you prove that ? I tried it many times and I get stuck

Well that is why I put a question mark at my statement of the
converse. I wondered if you asked what you meant to ask. Here is what

Post by Delta
How do you prove the converse of this theorem?
If a right triangle is inscribed in a semicircle then the hypoteneuse is the
diameter of the circle

So you meant to ask how to prove this theorem, not its converse.

--Lynn

[matt grime]

Post by Delta
How do you prove the converse of this theorem?
If a right triangle is inscribed in a semicircle then the hypoteneuse is the
diameter of the circle

looks like homework, so as a start, draw a line from the centre of the
circle to the apex of the triangle. should be straightforward from there.