Drag the red dot at the "@" mark to change the radius of the left circle
Drag the red dot at B to change the position of right circle
Type "r" to reset the diagram to its initial state
Main assertion: HE bisets the angle AHB.
Drag the red dot at the "@" mark to change the radius of the left circle
Drag the red dot at B to change the position of right circle
Type "r" to reset the diagram to its initial state
Newsgroup: sci.math From: rouben@pc18.math.umbc.edu (Rouben Rostamian) Subject: Re: A geometry question Date: Sat, 4 Jan 2003 00:55:04 +0000 (UTC) In article <gh7R9.114916$pe.4464622@news2.east.cox.net>, TCL <tlim1@cox.net> wrote: >In figure >I believe l_4 bisects angle O_1BO_2. >Is this true? >If so, how to prove it? Although it is said that a picture is worth a thousand words, pictures are inconvenient in Usenet's text-only medium. Therefore I will first translate your picture into words, then I will outline a solution. --- Statement of the problem ---------------------------------- Consider two circles with centers A and B and radii a and b. Let d be the distance between the centers. Assume d > a + b, that is, the circles are outside each other. Let a common tangent T1 to the two circles intersect the extended line AB at a point D outside the segment AB. Let another common tangent T2 to the two circles intersect the line segment AB at a point E between A and B. Let H be the foot of the perpendicular from D dropped onto T2. Show that HE bisects the angle H of the triangle AHB. --- Solution -------------------------------------------------- A straightforward computation shows that: EA = ad/(a+b) EB = bd/(a+b) HA = [a/(a-b)] sqrt(4ab-d^2) HB = [b/(a-b)] sqrt(4ab-d^2) whence EA/EB = HA/HB. See the Maple worksheet common_tangents.mws for details. Thus the line HE subdivides the side AB of the triangle AHB in proportion to the lengths of the sides HA and HB, proving that it is the bisector of the angle H. -- Rouben Rostamian <rostamian@umbc.edu>
Interesting side note. We have:
HD = 2ab/(a-b)
Therefore the length of the line segment HD is independent of d! Similarly, the distance of E from the tangent line T1 equals 2ab/(a+b) and is independent of d.
This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on January 3, 2003.