# Elementary Geometry, Plane and Solid; For Use in High Schools and Academies

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1901 edition. Excerpt: ...of a given line-segment is the least possible when it is bisected. 11. The sum of the squares on two internal segments of a given linesegment becomes greater as the point of section approaches one extremity. 12. Inscribe a square in a given semicircle. 13. If A, C, D, B are four points on a straight line so situated that D bisects CB, prove that the square on AC is less than the sum of the squares on AD and DB by twice the rectangle AD. DB. 14. If BAG is any acute angle and BD, CE, are drawn perpendicular to its boundaries AC, AB, respectively, show that the rectangle whose sides are equal to AB and AE is equal in area to the rectangle whose sides are equal to AC and AD. 15. If ABC be a right-angled triangle and CD be drawn perpendicular to the hypotenuse, then AD: DB = AC2: -BC3. Section II AREAS OF SIMILAR POLYGONS Proposition V 314. The areas of similar triangles are in the same ratio as the squares of any two homologous sides. 1. If the mid-points of two sides of a triangle are joined by a straight line, what part of the whole triangle is the smaller one so formed? Proposition VI 315. The areas of similar polygons are in the same ratio as the squares of any two homologous sides. B ED E' D' Let ABCDEF and A'B'CD'E'F be any two similar polygons of which AB and A'B' are homologous sides. It is required to prove that area of AB. F: area of A'B' F = AT?: A7!!'2. Proof. Divide the two polygons into triangles by drawing the diagonals from two homologous vertices, A and A'. These 316. Corollary. The areas of similar polygons are in the same ratio as the squares of any two homologous diagonals. Proposition VII 317. The area of the square described on the hypotenuse of a right triangle is equal to the sum of the areas of the squares described on...