15. The sum of the distances of any point in the base of an isosceles triangle from the equal sides is equal to the distance of either extremity of the base from the opposite side.
16. The three perpendiculars at the middle points of the sides of a triangle are concurrent. Hence prove that perpendiculars from the vertices on the opposite sides are concurrent [see Ex. 2].
17. Inscribe a lozenge in a triangle having for an angle one angle of the triangle.
18. Inscribe a square in a triangle having its base on a side of the triangle.
19. Find the locus of a point, the sum or the difference of whose distance from two fixed lines is equal to a given length.
20. The sum of the perpendiculars from any point in the interior of an equilateral triangle is equal to the perpendicular from any vertex on the opposite side.
21. The distance of the foot of the perpendicular from either extremity of the base of a triangle on the bisector of the vertical angle, from the middle point of the base, is equal to half the difference of the sides.
22. In the same case, if the bisector of the external vertical angle be taken, the distance will be equal to half the sum of the sides.
23. Find a point in one of the sides of a triangle such that the sum of the intercepts made by the other sides, on parallels drawn from the same point to these sides, may be equal to a given length.
24. If two angles have their legs respectively parallel, their bisectors are either parallel or perpendicular.