perplexus.info :: Geometry : Folded Paper

Folded Paper (Posted on 2008-04-29)

A square sheet of paper ABCD is folded with D falling on BC at D', with A falling on A', and A'D' intersecting AB at E.

Prove that the inradius of triangle EBD' is equal to |A'E|.

Submitted by Bractals

Rating: 4.0000 (1 votes)

Solution: (Hide)

Let F be the intersection of the fold and AB.

Let r and q be the inradii of the similar right triangles EA'F and EBD' respectively.
r(|EF| - |A'E|) = r|EF| - r|A'E| = q|ED'| - q|BE| = q(|A'D'| - |A'E|) - q(|BA| - |EF| - |FA|) = q(|A'D'| - |A'E|) - q(|A'D'| - |EF| - |FA'|) = q(|EF| + |FA'| - |A'E|) = ½(|FA'| + |A'E| - |EF|)(|FA'| - |A'E| + |EF|) = ½(|FA'|² - |A'E|² + 2|A'E||EF| - |EF|²) = ½(|FA'|² - |A'E|² + 2|A'E||EF| - |FA'|² - |A'E|²) = ½(2|A'E||EF| - 2|A'E|²) = |A'E|(|EF| - |A'E|) ===> r = |A'E|

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Puzzle Thoughts	K Sengupta	2024-02-24 00:14:54

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