Intersecting Chords Theorem
by Puzzled417, Apr 2, 2016, 10:43 PM
Theorem. Let a chord intersect a circle 
at points 
and another chord intersect the circle at 
distinct from . Denote by 
the angle formed between the two chords that subtends 
if they intersect. Then 
.
![[asy]
import graph; usepackage("amsmath"); size(6.6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.98,xmax=6.62,ymin=-2.94,ymax=6.52;
pair B=(0.96,0.14), A=(-2.8006041293639656,3.9917098915032883), D=(-2.234884510639143,0.10718758631715364), C=(1.9737458589080052,1.6565373027178305);
draw(circle((-0.66,2.32),2.7160265094435285)); draw(A--B); draw(D--C); label("$\angle{X}$",(-0.08,1.66),SE*lsf);
dot(B,linewidth(3.pt)+ds); label("$B$",(0.98,-0.2),NE*lsf); dot(A,linewidth(3.pt)+ds); label("$A$",(-3.,4.1),NE*lsf); dot(D,linewidth(3.pt)+ds); label("$D$",(-2.56,-0.2),NE*lsf); dot(C,linewidth(3.pt)+ds); label("$C$",(2.1,1.64),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](//latex.artofproblemsolving.com/5/b/7/5b7490a65875d00b838de556886d8dd3db0ac1b5.png)
at points 
and another chord intersect the circle at 
distinct from . Denote by 
the angle formed between the two chords that subtends 
if they intersect. Then 
.![[asy]
import graph; usepackage("amsmath"); size(6.6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.98,xmax=6.62,ymin=-2.94,ymax=6.52;
pair B=(0.96,0.14), A=(-2.8006041293639656,3.9917098915032883), D=(-2.234884510639143,0.10718758631715364), C=(1.9737458589080052,1.6565373027178305);
draw(circle((-0.66,2.32),2.7160265094435285)); draw(A--B); draw(D--C); label("$\angle{X}$",(-0.08,1.66),SE*lsf);
dot(B,linewidth(3.pt)+ds); label("$B$",(0.98,-0.2),NE*lsf); dot(A,linewidth(3.pt)+ds); label("$A$",(-3.,4.1),NE*lsf); dot(D,linewidth(3.pt)+ds); label("$D$",(-2.56,-0.2),NE*lsf); dot(C,linewidth(3.pt)+ds); label("$C$",(2.1,1.64),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/5/b/7/5b7490a65875d00b838de556886d8dd3db0ac1b5.png)
![[asy]
import graph; usepackage("amsmath"); size(6.6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.98,xmax=6.62,ymin=-2.94,ymax=6.52;
pair O=(-0.66,2.32), B=(0.96,0.14), A=(-2.8006041293639656,3.9917098915032883), D=(-2.234884510639143,0.10718758631715364), C=(1.9737458589080052,1.6565373027178305);
draw(circle(O,2.7160265094435285)); draw(A--B); draw(D--C); label("$\angle{X}$",(-0.08,1.66),SE*lsf); draw(A--C,linetype("2 2")); draw(D--B,linetype("2 2")); draw(A--O,linetype("2 2")); draw(O--C,linetype("2 2")); draw(O--B,linetype("2 2")); draw(O--D,linetype("2 2"));
dot(O,linewidth(3.pt)+ds); label("$O$",(-0.58,2.44),NE*lsf); dot(B,linewidth(3.pt)+ds); label("$B$",(0.98,-0.2),NE*lsf); dot(A,linewidth(3.pt)+ds); label("$A$",(-3.,4.1),NE*lsf); dot(D,linewidth(3.pt)+ds); label("$D$",(-2.56,-0.2),NE*lsf); dot(C,linewidth(3.pt)+ds); label("$C$",(2.1,1.64),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/f/2/5/f2505f30c9b5239de23160df230a704d4ecaa064.png)
and also that 
and 
.
and 
and thus upon adding both equations we get 
.
and 
,
.
This post has been edited 1 time. Last edited by Puzzled417, Apr 12, 2016, 3:33 PM
May 2016
April 2016