The discussion centers on determining the number of right-angled triangles that can be formed using 24 points, $A_1, A_2, \ldots, A_{24}$, equally spaced on the circumference of circle $O$. It is established that a triangle inscribed in a circle is right-angled if its hypotenuse is the diameter of the circle. Therefore, any three points that include the endpoints of a diameter will form a right-angled triangle. Given 24 points, there are 12 unique diameters, leading to 12 distinct right-angled triangles.
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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.