[數學]證明AN垂直NM

[數學]證明AN垂直NM

Raceleader 於 星期四 六月 19, 2003 12:23 pm


設ABC為三角形,D是過點A的高之垂足。設點E、F位於一條過點D的直線分別使得AE垂直BE且AF垂且CF ,並且E和F與D不同。設M、N分別為線段BC、EF的中點,證明AN垂直NM。

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Raceleader
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---- 於 星期四 六月 19, 2003 12:52 pm


angle AEB=angleADB=90
A,B,D,E are concyclic (converse of angle in same seg.)
angle ADC=angle AFC=90
A,D,C,F are concyclic (opp. angles supp.)

angle ABC=angle AEF (angle in same seg.)
angle ACB=angle AFE (angle in same seg.)
Angle BAC=180-angle ABC-angleACB(angle sum of triangle)
=180-angle AEF-angleAFE
=angle EAF
Therefore ABC~AEF (AAA)
angle ABM = angle AEN (corr. angle, ~tri.)
AB/AE = BC/EF =BM/EN (corr. sides, ~tri.)
Therefore ABM~AEN (ratio of two sides, included angle)
angle AMB = angle ANE (corr. angles, ~tri.)
Therefore A,D,M,N are concyclic (converse of angle in same seg.)
Angle ANM + angle ADC = 180 (opp. angle, cyclic quad.)
Angle ANM = 90

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hsiaod 於 星期六 九月 06, 2003 2:34 pm


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hsiaod
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