June 1st, 2018, 09:54 AM  #1 
Senior Member Joined: Jan 2012 Posts: 140 Thanks: 2  About property of triangle
Hi, If the circle with altitude AD of triangle ABC as diameter intersects AB & AC at the point E and F, how can we prove that: EF = $\displaystyle \frac{\Delta}{R}$ Thx. Last edited by skipjack; June 1st, 2018 at 04:57 PM. 
June 1st, 2018, 12:55 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,964 Thanks: 1148 Math Focus: Elementary mathematics and beyond 
Please explain your notation thoroughly. What is $\Delta$? What is $R$?

June 1st, 2018, 02:49 PM  #3 
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147 
Although there may be other meanings for these symbols, usually $\displaystyle \Delta =$ area of $\displaystyle \triangle ABC$ and $\displaystyle R =$ the circumradius of $\displaystyle \triangle ABC$. If this is what the OP means, then the statement is false.

June 1st, 2018, 04:58 PM  #4 
Senior Member Joined: Jan 2012 Posts: 140 Thanks: 2  Yes area for delta & circum radius is R, which are usual notations. If wrong, what EF should be equal to in terms of those two?

June 1st, 2018, 08:30 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,978 Thanks: 2229 
As $\displaystyle \small\triangle$ = AD$\cdot$BC/2 = (EF/sin A)BC/2 = EF(BC/sin A)/2 = EF(2R)/2, EF = $\small\triangle$/R.

June 1st, 2018, 10:02 PM  #6 
Senior Member Joined: Jan 2012 Posts: 140 Thanks: 2 
Plz only tell me how AD = EF / sin A. If posssible you can link me to a diagram.. Thx. 
June 2nd, 2018, 08:00 AM  #7 
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147 
Am I missing something? The area of the triangle is fixed and $\displaystyle R$ is fixed but as the diameter spins, $\displaystyle EF$ varies in length.

June 2nd, 2018, 10:35 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,978 Thanks: 2229 
The question specifies a circle with AD as diameter. That circle is the circumcircle of triangle AEF, not triangle ABC. As AD is a diameter of the circumcircle of triangle AEF, AD = EF/sin A is a consequence of the sine rule. In the diagram below, the circumcircle of triangle ABC isn't shown, but its radius is R. TriangleProperty.PNG 
June 2nd, 2018, 04:15 PM  #9 
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147 
Ahh ... I misread the problem!

June 3rd, 2018, 04:23 AM  #10  
Senior Member Joined: Jan 2012 Posts: 140 Thanks: 2  Quote:
Last edited by skipjack; July 23rd, 2018 at 01:05 PM.  

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