To Find The Length For A Rectangle Whose Area Shall Be Equal To That Of A Given Square, The Breadth Of The Rectangle Being Also Given

Let ABCD be the given square, and DE the given breadth of rectangle; continue the line BC to F, and draw the line DF; also, continue the line DC to g, and draw the line Ag parallel to DF; from the intersection of the lines at g, draw the line gd parallel to DE, and Erf parallel to Dg; then EDdg is the rectangle as required.

To Bisect Any Given Triangle

Supose ABC the given triangle; bisect one of its sides, as AB in e, from which describe the semicircle ArB; bisect the same in r, and from B, with the distance Br, cut the diameter AB in v; draw the line vy parallel to AC, which will bisect the triangle as required.

To Describe A Circle Of Greatest Diameter In A Given Triangle

Bisect the angles A and B, and draw the intersecting lines AD, BD, cutting each other in D; then from D as centre, with the distance or radii DC, describe the circle Cef, as required.

To Form A Rectangle Of Greatest Surface In A Given Triangle

Let ABC be the given triangle; bisect any two of its sides, as AB, BC, in e and d; draw the line ed; also at right angles with the line ed, draw the lines ep,dp, and eppd is the rectangle required.