Angles Bisected
Source:
February 7, 2009
Problem Statement
Given triangle with bisecting , extended to and a right angle, then:
[asy]path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false)
{
pair M,N;
path mark;
M=t*0.03*unit(A-B)+B;
N=t*0.03*unit(C-B)+B;
if(flip)
mark=Arc(B,t*0.03,degrees(C-B)-360,degrees(A-B));
else
mark=Arc(B,t*0.03,degrees(A-B),degrees(C-B));
return mark;
}unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));pair P=(0,0), R=(3,2), Q=(4,0);
pair S0=bisectorpoint(P,R,Q);
pair Sp=extension(P,Q,S0,R);
pair D0=bisectorpoint(R,Sp), Np=midpoint(R--Sp);
pair D=extension(Np,D0,P,Q), M=extension(Np,D0,P,R);draw(P--R--Q);
draw(R--Sp);
draw(P--D--M);draw(anglemark2(Sp,P,R,17));
label("",P+(0.35,0.1));
draw(anglemark2(R,Q,P,11));
label("",Q+(-0.17,0.1));
draw(anglemark2(R,Np,D,8,true));
label("",Np+(+0.12,0.07));
draw(anglemark2(R,M,D,13,true));
label("",M+(+0.25,0.03));
draw(anglemark2(M,D,P,29));
label("",D+(-0.75,0.095));pen f=fontsize(10pt);
label("",R,N,f);
label("",P,S,f);
label("",Sp,S,f);
label("",Q,S,f);
label("",D,S,f);[/asy] (A)\ \angle m \equal{} \frac {1}{2}(\angle p \minus{} \angle q) \qquad (B)\ \angle m \equal{} \frac {1}{2}(\angle p \plus{} \angle q)
(C)\ \angle d \equal{} \frac {1}{2} (\angle q \plus{} \angle p) \qquad (D)\ \angle d \equal{} \frac {1}{2}\angle m \qquad (E)\ \text{none of these is correct}