a relation which cannot be true unless P and Q coincide with B.

14. In every triangle the intersection of the three altitudes, the intersection of the three medians, and the intersection of the three perpendiculars erected at the middle points of the sides, lie in a straight line; and the distance between the first two points is double the distance between the last two.

Let D, E, F (Fig. 52) be the three points in question. The line MN, joining the middle points of AB, BC, is I to AC. Hence, the A ADC and MNF are similar, and

Hence, the AADE and NEF are similar, and ZAED = 2 NEF; then DEF is a straight line. Also, from the AADE, NEF,

15. Two circles cut in point P. Through P three lines are drawn, meeting one of the circles in A, B, C, the other in D, E, F, respectively. Prove that the triangles ABC, DEF are similar.

16. In every triangle the product of two sides is equal to the product of the diameter of the circumscribed circle and the altitude upon the third side.

If AC, BC are taken as the two sides, draw the diameter CE, and join EB.

17. In every inscribed quadrilateral the product of the diagonals is equal to the sum of the products of the opposite sides.

Let ABCD be the quadrilateral. In AC take a point E such that LEDC ZADB. ▲ ADB and CDE are similar; also, the ▲ BCD and ADE. From these triangles obtain equations involving the sides, and then add them.

18. In every triangle the product of two sides is equal to the square of the bisector of the included angle plus the product of the segments into which it divides the third side.

If AC, BC are the two sides, and CD the bisector, produce CD to meet the circumscribed circle in E, and join BE. A ACD and ECB are similar; also, see No. 164.

19. In every triangle the sum of the squares of two sides is equal to twice the square of half the third side plus twice the square of the median drawn to the third side. (Nos. 162 and 163.)

20. In every triangle the difference of the squares of two sides is equal to twice the product of the third side and the projection of the median upon the third side.

21. In the diameter of a circle two points A, B are taken equally distant from the centre, and joined to a point Pin the circumference. Prove that the sum AP2+BP2 is constant for all positions of P. (Ex. 19.)

22. The sum of the squares of the sides of a parallelogram is equal to the sum of the squares of the diagonals (Ex. 19.)

23. The sum of the squares of the sides of any quadrilateral 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. (Ex. 19.)

24. The sum of the squares of the diagonals of a trapezoid is equal to the sum of the squares of the legs plus twice the product of the bases. (Ex. 23 and § 2, Ex. 56.)

25. Two triangles are similar if their sides are parallel each to each. Which are homologous sides?

26. Two triangles are similar if their sides are perpendicular each to each. Which are homologous sides?

27. If two similar triangles ABC, DEFhave their homologous sides parallel, the lines AD, BE, CF which join their homologous vertices meet in the same point.

Let AD and CF (Fig. 53) meet in O, BE and CF in Q. From the similar triangles in the figure, prove that

28. Two polygons are similar if their homologous sides are parallel each to each.

29. If two similar polygons have their homologous sides parallel, the lines joining their homologous vertices meet in the same point.

Let two of the lines AA', BB' meet in O, and let AA' meet any third line, as DD', in Q. Draw diagonals, and prove, as in Ex. 24, that must coincide with O.

The ratio AO: A'O, BO: B'O, etc., are each equal to the ratio of any two homologous sides.

The point O is called the centre of similitude of the polygons. If the homologous sides are directed the same way (Fig. 54), the two polygons are said to be similarly placed, and O is called the direct centre of similitude. If the homologous sides are directed opposite ways (Fig. 55), the two polygons are said to be inversely placed, and O is called the inverse centre of similitude.

30. If a point O is joined to the vertices of a polygon ABCDE, and upon the lines OA, OB, etc., the lengths OA', OB', etc., are laid off, so that the ratios OA': OA, OB': OB, etc., are equal, the polygon A'B'C'D'E' is similar to the polygon ABCDE.

31. If a point O is joined to the vertices of a polygon ABCDE, and through any point A' in OA a line parallel to AB is drawn, meeting OB in B', and through B' a line parallel to BC, meeting OC' in C', etc., the polygon A'B'C' D'E' is similar to the polygon ABCDE.

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DEFINITIONS. Let AB be a given line, m:n a given ratio, Pa point in AB, such that PA: PB = m: n; then AB is said to be divided internally or externally in the ratio m:n, according as Pis between A and B or in AB produced.

If the line joining the centres of two circles is divided externally and internally in the ratio of their radii, the points of division are called the direct and the inverse centres of similitude, respectively, of the two circles.

It follows from these definitions that the point of contact of two circles which touch externally is an inverse centre of similitude; and that the point of contact of two circles, one of which touches the other internally, is a direct centre of similitude.

32. The line joining the extremities of two parallel radii of two circles passes through the direct centre of similitude, if the radii have the same direction; and through the inverse centre if the radii have opposite directions.

Let r, rꞌ (Fig. 56) denote the radii of the circles, O, O' their centres; and let OM be to OR, ON O'T, RM and OƠ′ meet in P. NT and OO meet in Q. Then, OP: O'P=r: r', and OQ: O'Q=r: r. . Pis the direct, and Q the inverse, centre of similitude.