Proposed Problem

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

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Complete Problem 400

Level: High School, SAT Prep, College geometry

## Tuesday, December 8, 2009

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

Labels:
angle bisector,
circumcircle,
congruence,
perpendicular,
triangle

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|OG|=|OH|

ReplyDelete|OC|=|OD|=r

|GC|=|DH|

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

ReplyDeletesuggest to others

ReplyDelete1) 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

reason for step 1:

ReplyDeletehttp://i49.tinypic.com/69omdj.gif

and

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

Thanks!

ReplyDeleteW/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.

ReplyDeleteAjit

step 1 and 2 need third condition of congruence,

ReplyDeleteangle 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

Triangles HFC and EGC are similar

ReplyDeleteSo OH = OG

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

Sumith Peiris

Moratuwa

Sri Lanka

OGH isosceles ==>

ReplyDeleteperpendicular from O to CD bisect GH, and also bisect CD ==>

GC = DH

More or less the same as the last few but with a few extra steps included

ReplyDelete1. 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.