###### Solutions and Notes to Supplementary Problems

89 SOLVED PROBLEMS

Question: P.417b

## Given a quadrilateral with sides a, b, c, d, diagonals e, f, and area S, we have 4e² f² = (a² + c² − b² − d²)² + 16S ². The angle V of the diagonals is given by tan V = 4S/a² + c² – b² – d². Deduce from this a solution of the preceding exercise. (Having fixed the position of one side, each of the ...

We use figure t417a, letting AB = a, BC = b, CD = ...
Question: P.420

## The radii of the circles circumscribing (Exercise 66) the quadrilaterals determined by the bisectors of the interior (or exterior) angles of a quadrilat-eral are in the ratio a+c-b-d/a+c+b+d, where a, b, c, d are the sides of the given quadrilateral, taken in their natural order. ...

Suppose the given quadrilateral is MNPQ, and suppo...
Question: P.419

## Among all the closed curves of same length, the circle is the one whose interior has the largest area. (Consider the ratio S/p² for a polygon inscribed in a circle and a polygon inscribed in a curve of the same length, the number of sides being the same in the two cases, and let the number of sides ...

We consider a circle with circumference C, and a c...
Question: P.418b

## Among all polygons with the same number of sides and the same perimeter, the largest is the regular polygon. (Assuming that a polygon of maximum area exists, we can use the preceding exercises and Exercise 331 to show that this polygon must be regular.) The result can be restated as follows: If S ...

Suppose ABCDE... is the polygon referred to, with ...
Question: P.418

## Construct a cyclic quadrilateral knowing its sides. ...

Suppose (fig. t418) ABCD is the required quadrilat...
Question: P.416

## Construct a triangle knowing a side, the perimeter, and the area (construct the figure formed by the inscribed circle and an escribed circle). Among all triangles with a given side and given area, which has the smallest perimeter? Among all triangles with a given side and given perimeter, which has ...

The solution to this problem is only slightly diff...
Question: P.415

## Construct a triangle knowing an angle, the perimeter, and the area (Exercises 90b, 299). Among all triangles with a given angle and given perimeter, which one has largest area? ...

Suppose we know the measure of angle \hat{{...
Question: P.414

## Another solution to Exercise 329: to draw through a given point inside an angle a secant which forms, with the sides of the angle, a triangle with given area. Construct first the parallelogram with a vertex at the given point, and two sides on the sides of the angle. This parallelogram cuts from ...

Suppose (fig. t414) we must draw secant MON throug...
Question: P.413b