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Mathematics LibreTexts

6.3: Method of similar triangles

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The proof of the Pythagorean theorem given above uses the method of similar triangles. To apply this method, one has to search for pairs of similar triangles and then use the proportionality of corresponding sides and/or equalities of corresponding angles. Finding such pairs might be tricky at first.

Exercise 6.3.1

Let ABC be a nondegenerate triangle and the points X,Y, and Z as on the diagram. Assume CAYXBC. Find four pairs of similar triangles with these six points as the vertexes and prove their similarity.

截屏2021-02-08 下午1.39.11.png

Hint

By the AA similarity condition (Theorem 6.1.2), AYCBXC. Conclude that YCAC=XCBC. Apply the SAS similarity condition to show that ABCYXC.

Similarly, apply AA and equality of vertical angles to prove that AZXBZY and use SAS to show that ABZYXZ.


This page titled 6.3: Method of similar triangles is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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