352. Exercise.-If from one of the acute angles of a right-angled triangle a straight line be drawn bisecting the opposite side, the square of that line will be less than the square of the hypotenuse by three times the square of half the side bisected.

353. Exercise.-If two circles intersect each other, the tangents drawn from any point of their common chord produced are equal. 8321

354. Exercise. If two circles intersect each other, their common chord if produced will bisect their common tangent. $321

355. Exercise.-I. The sum of the squares of two sides of a triangle is equal to twice the square of half the third side, plus twice the square of the median drawn to the third side.

II. The difference of the squares of two sides of a triangle is equal to twice the product of the third side by the projection of the median upon the third side.

Hint.-The median BD divides ABC into two triangles, one acute angled and the other obtuse angled (provided AB and BC are not equal). Apply SS 325, 326.

356. Exercise. In any quadrilateral the sum of the squares of the four sides is equal to the sum of the squares of the diagonals plus four times the square of the line joining the middle points of the diagonals.

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Hint.-Apply § 355, I. to the triangles ABC, ADC. and BED, and combine equations thus obtained.

357. Exercise.-The product of two sides of a triangle is equal to the product of the diameter of the circumscribed circle and the altitude upon the third side.

M

Hint.-Let ABC be the triangle. Draw the altitude BD and the diameter BM. Prove the triangles BAM and BDC similar. SS 201, 202, 276 358. Exercise.-In an inscribed quadrilateral, ABCD, if F is the intersection of the diagonals AC and BD, then

Hint.

In the triangles ABD and CBD, draw the altitudes AM and CN

and apply $357. Then compare triangles AFM and CFN.

359. Exercise.-The product of two sides of a triangle is equal to the square of the bisector of their included angle plus the product of the segments of the third side formed by the bisector.

Hint.-Circumscribe a circle about ABC and produce the bisector to cut the circumference in M. Prove the triangles ABD and MBC similar. Apply § 320.

PROBLEMS OF CONSTRUCTION

360. Exercise.—To produce a given straight line MN to a point X, such that MN: MX=3:7.

361. Exercise.-To construct two straight lines having given their sum and ratio.

362. Exercise.-Having given the lesser segment of a straight line divided in extreme and mean ratio, to construct the whole line.

363. Exercise.-To construct a triangle having a given perimeter and similar to a given triangle.

364. Exercise.-To construct a right triangle having given an acute angle and the perimeter.

365. Exercise. To divide one side of a given triangle into segments proportional to the other two sides.

366. Exercise.-In a given circle to inscribe a triangle similar to a given triangle.

367. Exercise.-About a given circle to circumscribe a triangle similar to a given triangle.

368. Exercise.-To inscribe a square in a semicircle.

Hint.-At B draw CB equal and perpendicular to the diameter. Join OC cutting the circumference in M, and draw MD parallel to CB. Prove MD the side of the required square by § 275.

369. Exercise.-To inscribe a square in a given triangle.

Hint.

On the altitude AD construct the square ADFE and draw BE cutting the side AC at M. From M draw MN and MP parallel to EF and AE respectively. Prove these lines equal and sides of the required square. 370. Exercise.-To inscribe in a given triangle a rectangle similar to a given rectangle.

371. Exercise.-To inscribe in a given triangle a parallelogram similar to a given parallelogram.

372. Exercise.-To construct a circumference which shall

pass through two given points and be tangent to a given straight line.

Hint.-Let AB be the given line, P and P' the points.

If the straight line PP' is parallel to AB, the solution is simple. If PP' is not parallel to AB, it will cut it at some point X, and the distance from X to Y, the required point of tangency, may be determined by § 321.

PROBLEMS FOR COMPUTATION

373. (1.) In the triangle ABC, DE is drawn parallel to , BC= 56, and AE=24, find AC and DE.

BC. If

AD 4

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(2.) The sides of a triangle are 3, 5, and 7. In a similar triangle the side homologous to 5 is equal to 65. Find the other two sides of the second triangle.

(3.), The shadow cast upon level ground by a certain church steeple is 27 yds. long, while at the same time that of a vertical rod 5 ft. high is 3 ft. long. Find the height of the steeple.