April 8th, 2018, 11:12 PM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 470 Thanks: 1  How angle ACB is 90?
In circle ABC where O is the center, there is a triangle ABC. but how angle ACB is 90? 
April 8th, 2018, 11:52 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond 
That result is known as Thales's theorem.

April 9th, 2018, 03:49 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
A more general theorem is that if an angle is inscribed in a circle (all three vertices on the circle) then the angle measure is half the measure of the arc it cuts off. Since AB goes through the center, O, it is a diameter. What is the measure of the arc AB? So what is the angle ACB?

April 9th, 2018, 03:50 AM  #4 
Member Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1 
The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference of that circle. The angle AOB is twice the angle ACB. ******************** In our case, the angle AOB = 180 degrees, so the angle ACB must be half of the angle AOB = (1/2)*(180) = 90 degrees. 
April 9th, 2018, 06:51 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,969 Thanks: 2218 
As triangles AOC and BOC are isosceles, $\small\angle$ACB = $\small\angle$ACO + $\small\angle$BCO = $\small\angle$CAO + $\small\angle$CBO. Hence $\small\angle$ACB is half the sum of the internal angles of triangle ABC. 
April 9th, 2018, 07:54 AM  #6 
Senior Member Joined: Aug 2014 From: India Posts: 470 Thanks: 1  
April 9th, 2018, 10:10 PM  #7 
Member Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1  
April 16th, 2018, 06:48 AM  #8 
Newbie Joined: Mar 2018 From: Washington Posts: 6 Thanks: 0 Math Focus: Geometry 
Is it obvious to me alone? Using Thales's theorem is all easy. Usually if I do not understand something, I'll first read all the relevant information, if I have not found my answer, then I ask other sources starting from mathforums ending with help sites in homework. Thanks to this, I now feel much more strongly in various fields of activity. Learn while young. Last edited by skipjack; April 16th, 2018 at 07:12 AM. 
April 16th, 2018, 08:01 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
Scotty, can we buy you a oneway ticket to another site?

April 16th, 2018, 09:45 AM  #10 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Behave, Denis!


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