Midpoint theorem (triangle): Difference between revisions

Revision as of 08:45, 31 August 2022

Revision as of 04:00, 15 August 2023

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The triangle formed by the three parallel lines through the three midpoints of sides of a triangle is called its [[medial triangle]].

==Proof==

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|style="width:800px" valign="top" |[[File:Midpoint Theorem proof.png|thumb|304x304px]]

'''Given''': In a <math>\triangle ABC </math> the points M and N are the midpoints of the sides AB and AC respectively.

'''[[Geometric Construction|Construction]]''': MN extended to D where MN=DN, join C to D.

'''To Prove''':

*<math>MN\parallel BC</math>

*<math>MN={1\over 2}BC</math>

'''Proof''':

*<math>AN=CN</math> (given)

*<math>\angle ANM=\angle CND</math> (vertically opposite angle)

*<math>MN=DN</math> (constructible)

Hence by [[Side angle side]].

:<math>\triangle AMN\cong\triangle CDN </math>

Therefore, the corresponding sides and angles of congruent triangles are equal

*<math>AM=BM=CD</math>

*<math>\angle MAN=\angle DCN</math>

[[Transversal (geometry)|Transversal]] AC intersects the lines AB and CD and alternate angles ∠MAN and ∠DCN are equal. Therefore

*<math>AM\parallel CD\parallel BM</math>

Hence BCDM is a [[parallelogram]]. BC and DM are also equal and parallel.

*<math>MN\parallel BC</math>

*<math>MN={1\over 2}MD={1\over 2}BC</math>

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==References==