The discussion revolves around determining how many triangles formed by 24 points equally spaced on the circumference of a circle can be classified as right-angled. The focus is on the geometric properties of inscribed triangles and the conditions under which they can be right-angled.
Participants generally agree on the geometric principle that a right-angled triangle inscribed in a circle must have its hypotenuse as the diameter. However, the exact number of such triangles remains unaddressed, indicating that the discussion is not resolved.
The discussion does not specify the total number of right-angled triangles that can be formed, nor does it explore the combinatorial aspects of selecting points from the 24 available.
Albert said:24 points $A_1,A_2,-----,A_{24}$ equally divide the circumference of circle $O$
,any three of the 24 points will determine an inscribed triangle,
now how many triangles can be proclaimed as right angled ?
HallsofIvy said:I think you mean "diameter" rather than "diagonal". Albert, the measure of an angle inscribed in a circle is 1/2 the measure of the arc it subtends. In order that the angle be right, that arc subscribed must be a semicircle so that the hypotenuse of the triangle is a diameter of the circle.
HallsofIvy said:I think you mean "diameter" rather than "diagonal". Albert, the measure of an angle inscribed in a circle is 1/2 the measure of the arc it subtends. In order that the angle be right, that arc subscribed must be a semicircle so that the hypotenuse of the triangle is a diameter of the circle.