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In the above construction, the tangent theorem has been used to construct the mean proportional, but any other construction of a mean proportional may be used.

Ex. 1000. Give a purely geometric proof of Ex. 999.

Ex. 1001. In a circle O a diameter AB = 2 r is drawn. From the midpoint C of the semicircle AB to draw a chord CD, meeting AB in E, so that DE shall be equal to a given line p.

Analysis. Let CE = x, OE = y.

and

x2 = y2 + r2,

x.p=(r+y) (r — y).

Eliminate y and solve. Then

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Ex. 1002. Given lines AB and p. In the line AB, to find a point C so that AC2 - CB2 = p2.

Ex. 1003. In the line AB, to find a point C so that AC2 = 2 CB2.

Ex. 1004. In the median AD of ▲ ABC, to find a point X such that XD and the perpendiculars dropped from X upon AB and AC divide the figure into three equivalent parts.

Ex. 1005. In a given square ABCD, to inscribe an equilateral triangle, having one vertex at A.

Ex. 1006. From a point P without a circumference, to draw a secant which is bisected by the circumference.

Ex. 1007. In a given square, to inscribe another square, having a given side.

Ex. 1008. To inscribe a square in a semicircle.

Ex. 1009. In the triangle ABC, to inscribe a parallelogram having a given area, and having an angle common with ▲ ABC.

Ex. 1010. From the midpoint A of arc BC, to draw a chord AD, intersecting chord BC in E, so that DE is equal to a given A line p.

Ex. 1011. To construct a triangle, having given the base, the vertical angle, and the bisector of that angle. (Ex. 1010.)

B

E

D

C

Ex. 1012. Upon a given line as hypotenuse to construct a right triangle one of whose arms is a mean proportional between the other arm and the hypotenuse.

Ex. 1013. To transform a given square into a rectangle having a perimeter equal to twice the perimeter of the given square.

Ex. 1014. Given two concentric circles. To draw a chord in the larger circle so that it equals twice the chord formed in the smaller circle.

MAXIMA AND MINIMA OF PLANE FIGURES

439. DEF. A maximum is the greatest of all magnitudes of the same kind; a minimum, the smallest.

440. DEF. Isoperimetric figures are those which have equal perimeters.

PROPOSITION I. THEOREM

441. Of all triangles having given two sides, that in which those two sides are perpendicular to each other

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Hyp. In ▲ ABC and DFE, AB=DF, AC=DE, ≤ A=rt. ≤,

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.. ▲ ABC and DEF have equal bases and unequal altitudes.

...

▲ ABC A DEF.

: AABC is a maximum.

(Why?)

Q.E.D.

Ex. 1015. To divide a given line into two parts so that the rectangle contained by the segments is a maximum.

Ex. 1016. In the hypotenuse of a right triangle to find a point so that the sum of the squares of perpendiculars drawn from the point upon the arms is a minimum.

PROPOSITION II. THEOREM

442. Of all triangles having the same base and equal areas, the isosceles triangle has the minimum perimeter.

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