Baccalauréat, toutes séries Session de juin 2010

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Baccalauréat, toutes séries Session de juin 2010

Épreuve de section européenne

The Feuerbah Cirle

In everytriangle, the threemidpoints of the sides, the three basepoints ofthe altitudes, and

themidpointsofthe three altitude setionstouhingthe verties lieon airle.

b

A

b B ^b C ^b

H

b

K

b

L

b O

b

C ^′

b B ^′

b

A ^′

b

b b b

b b

Theproofonsistsoftwosteps:intherstwedemonstratethattheirleirumsribingthetriangle

ofthethreemidpointsofthesidespassesthroughthebasepointsofthealtitudes;andintheseond we

showthattheirleirumsribingthetriangleofthealtitudebasepointspassesthroughthemidpoints

ofthealtitudesetions.

Step I. Let

A ′

,

B ′

and

C ′

represent themidpoints,respetively, ofsides

BC

,

AC

and

AB

. Let

H

be thebasepointof thealtitude

AH

. Thenthetrapezoid

1

H A ′ B ′ C ′

isisoselesand itisthereforea

quadrilateralinsribedinairle,thatwewillname

C

.Inthesamemannerwewoulddemonstrate theotheraltitudesbasepoints,namely

K

and

L

,lieonirle

C

, irumsribingtriangle

A ′ B ′ C ′

.

Step II. Letthealtitudesof thetriangle

ABC

be

AH

,

BK

,

CL

,and

O

theirpointofintersetion.We willnowshowthattheenterofeahaltitudesetiontouhingavertex,letussaysetion

OC

,also liesonirle

C

.[Endof thisproof tobeompletedin question 3.℄

AdaptedfromHeinrihDörrie's100GreatProblems ofElementary Mathematis,Dover,1965.

Questions

1. Whatarethemidpointsofthethreealtitudesetionstouhingtheverties?

2. InstepI,explainwhy

H A ′ B ′ C ′

isatrapezoid andwhyitisisoseles[Hint :

H C ′

istheradius of

theirlewhihhas

AB

asitsdiameter.℄

3. CompletingstepII:

(a) Considertriangle

OBC

:what areitsaltitudebases?Whatistheirirumsribedirle? (b) ApplytheresultofstepI totriangle

OBC

:whatistheirumirleofthese3points? 4. Showthatanisoselestrapezoidisalwaysinsribedinairle(youanskeththesituation;itmay

beusefultoonsidertheperpendiularbisetorsofthetwonon-parallelsides.)

2010-01TheFeuerbahCirle

Baccalauréat, toutes séries Session de juin 2010