## Tuesday, December 8, 2009

### Problem 400. Triangle, Angle bisector, Circumcircle, Perpendicular, Congruence

Proposed Problem
Click the figure below to see the complete problem 400 about Triangle, Angle bisector, Circumcircle, Perpendicular, Congruence.

Complete Problem 400
Level: High School, SAT Prep, College geometry

1. |OG|=|OH|
|OC|=|OD|=r
|GC|=|DH|

2. Reasons for step 1: |OG|=|OH| ?

3. suggest to others

1) draw HP perpendicular to AC ( P on AC )
2) draw OK perpendicular to HG ( K on HG )

about first comment: step 1 and step 2 are not enough for step 3

4. reason for step 1:
http://i49.tinypic.com/69omdj.gif

and
step 1 and step 2 are enough for step 3, i think

5. W/o having to refer to any other figure or any construction, we can easily see that quad. OFCE is concyclic and hence ang. HOG = ang. C while ang. OGH = ang. EGC = 90-C/2. Therefore, ang. OHG = 180 – C - (90-C/2)= 90 – C/2 or ang. OHG = ang. OGH. Hence etc.
Ajit

6. step 1 and 2 need third condition of congruence,
angle DOH = angle COG ? verify please

my solution
to draw OK perpendicular to HG give
1) OK is median => HK = KG
2) OK is diameter perpendicular to chord DC => DK=KC
so DH=DK-HK
and GC=KC-KG

7. Triangles HFC and EGC are similar

So OH = OG

So Tr.s ODH and OGC are congruent ASA and the result follows

Sumith Peiris
Moratuwa
Sri Lanka

8. OGH isosceles ==>
perpendicular from O to CD bisect GH, and also bisect CD ==>
GC = DH

9. More or less the same as the last few but with a few extra steps included
1. Right triangle CEG is similar to right triangle CFH because of the angle bisector.
2. So angle CGE = angle HGO = angle FHC and triangle OHG is isosceles.
3. OD = OC since they are radii.
4. So CDO is isoscleses and angle ODC = OCD.
5. Angle OHD = OGC since they are supplementary to 2 congruent angles.
6. We now have 2 out of 3 angles and 2 out of 3 sides congruent in ODH and OGC which is more than enough to infer a missing angle and use either SAS or ASA to show the triangles are congruent.
7. So DH = CG.