## Wednesday, February 18, 2009

### Problem 14: Triangle, Cevian, Midpoint, Angles

Proposed Problem

See complete Problem 14 at:
http://gogeometry.com/problem/problem014.htm

Level: High School, SAT Prep, College geometry

1. WLOG one may assume D:(0,0), A:(-1,0), C:(1,0). Let angle BCA = m and angle BAC=angle CBD = 2m
Thus angle BDA = 3m. Further, let tan(m)=t
Now, DB: y =(-tan3m)x =-x(3t-t^3)/(1-3t^2)
CB:y=(1-x)t which give B as((3t^2-1)/2(t2+1),(3t-t^3)/2(t2+1)) while A is (-1,0). hence slope BA=
(3t-t^3)/(5t^2+1) which as we know is tan(2m)=2t/(1-t^2). The equation obtained: (3t-t^3)/(5t^2+1)=2t/(1-t^2) has five solutions of which the only admissible solution is t=0.2679491924 which gives m=15 deg. Thus, angle BCA = 15 deg.
Alternatively, in triangle ABD, AD/BD =sin(5m)/sin(2m) while triangle BDC gives, CD/BD=sin(2m)/sin(m). Since AD=CD, we can say that sin(5m)/sin(2m)=sin(2m)/sin(m) which can be solved to get m=15 deg as before.
Now, having proven my bonafides, can someone please show me how such problems can be solved more elegantly by plane geometry alone?
Ajit: ajitathle@gmail.com

2. http://geometri-problemleri.blogspot.com/2009/05/problem-21-ve-cozumu.html

3. http://ahmetelmas.files.wordpress.com/2010/05/cozumlu-ornekler.pdf

5. CE perpendicular to AB and BE=EH. So , angle AHC=3x. Because, angle BDA=3x ,
BDCH , is a cyclic quadrilateral , therefore , angle BHD=x. The triangle AEC, is rectangular and AD=DC. So, ED=DA=DC, and angle BED=2x, therefore, angle EDH=x. So ,DE=EH=EB .Then angle BDH=90 degrees and 3x+2x+x=90 .Therefore x=15 degrees.
image : http://img248.imageshack.us/img248/8017/geogebra4.png

1. Excellent solution Michael

7. Assume length of AC=2
Note triangle ABC and BDC are similar.
AC/BC=BC/DC
2/BC=BC/1
Hence BC=√2

Construct point E which lies on BC and <EDC=90. By joining AE, triangle EBA is again similar to ABC and BDC, and EA=EC.Let length of AE and EC=a
AC/BC=AE/AB
2/√2=AE/AB
AB=a/√2

BC/AB=AB/BE
√2/AB=AB/BE
BE=AB^2/√2
BE=(a/√2)^2/√2
BE=(√2)a^2/4

BE+EC=√2
(√2)a^2/4 + a - √2=0 (quadratic equation)
a=1.03527618
In triangle EDC,
cos x=1/a
x=15