User Name Remember Me? Password

 Trigonometry Trigonometry Math Forum

 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

Quote:
 Originally Posted by mrtwhs 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.
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.
Attached Images Pic1.jpg (18.5 KB, 0 views) 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:
 Originally Posted by skipjack 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. Attachment 9736
Ohk.. That's where I was confused too... It basically meant that R = cicumradius of triangle AEF (not that of ABC).

Last edited by skipjack; July 23rd, 2018 at 01:05 PM. Tags property, triangle Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post WWRtelescoping Complex Analysis 2 January 27th, 2014 03:38 AM usermind Calculus 2 May 27th, 2012 03:43 PM Tensor Field Advanced Statistics 1 February 11th, 2011 01:42 PM sameapple Algebra 2 September 27th, 2009 05:00 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      