PROBLEM VII.
TO inscribe the greatest square possible in a given Triangle, i. e. having given the heighth of the Triangle CD (Fig. 20) and the Base AB, to find a portion of the altitude CE, which being cut off there shall remain ED=FG.
Make the base AB=a; the altitude CD=b, CE=x; then will ED or FG=b−x.
By reason of the similitude of the Triangles ABC and FGC you'l have as AB to CD so FG to CE. 〈 math 〉〈 math 〉
Therefore the Rectangles of the means and extremes will be equal, i. e. ax=bb−bx; and adding on both sides bx, ax+bx=bb, and dividing by 〈 math 〉〈 math 〉.
Construction. Upon the side of the Triangle CB produced, make CH=b, and HI=a, so that the whole Line shall be a+b. And having joined ID and parallel to it HE drawn from H, the part CE will be cut off, which is that sought.
For as CI to CD so CH to CE, 〈 math 〉〈 math 〉 according to the second case of simple Effections.
Arithmetical Rule. Square the given heighth of the Tri∣angle, and divide the Product by the sum of the base and al∣titude; and the quote is the part to be cut off CE. E. g. sup∣pose CD=10, and AB=15.