MHB How many triangles can be proclaimed as right angled ?

[Albert1]

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 ?

 

[kaliprasad]

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 ?

Spoiler

$A_1,A_{13}$ lie on on a diameter l so $A_1,A_{13}$ with any of other A that is 22 values form a right angled triangle

there are 12 diameters so number of right angled triangles = 22 * 12 = 264

 

[HallsofIvy]

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.

 

[Albert1]

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.

Spoiler

diagonal $\overline{A_1A_{13}}$ happens to be a diameter of the given circle

 

[kaliprasad]

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.

Spoiler

Thanks. done the correction in line