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Notes on Euclid I., 47.

Published online by Cambridge University Press:  20 January 2009

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ABC is a triangle, right-angled at A, and X, Y, Z are the centres of the squares described on the sides opposite the angles A, B, C; XD, XM, XN are respectively perpendicular to BC, CA, AB; MY, DY are joined, and DY meets AC at E; NZ, DZ are joined, and DZ meets AB at F.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1885

References

page 18 note * My attention was drawn, after the present paper was in proof, to an anticipation of the above property in the Educational Times, for August 1879, (Vol. 32, p. 241), where the mathematical editor, Mr J. C. Miller, proposes and solves the following proposition:

A square ia constructed on the hypotenuse of a right-angled triangle; prove that the distance of the middle of this square from each of the sides that contain the right angle of the triangle is equal to half the difference, or to half the sum, of these sides, according as the triangle and the square are on the same or on opposite sides of the hypotenuse.

page 20 note * From this we get a geometrical proof of the result Δ=s(sa)=(sb)(sc)for the area of a right-angled triangle.

For, with the usual notation, the result of (13) becomes

which give the two expressions for the area.