23. If there be four magnitudes such that the first is to the second as the third is to the fourth; then shall the first together with the second be to the excess of the first above the second as the third together with the fourth is to the excess of the third above the fourth.
For, the first together with the second is to the second as the third together with the fourth is to the fourth (V. 18). Therefore, alternately, the first together with the second is to the third together with the fourth as the second is to the fourth (V. 16).
Similarly, by V. 17 and V. 16, the excess of the first above the second is to the excess of the third above the fourth as the second is to the fourth.
Therefore, by V. 11, the first together with the second is to the excess of the first above the second as the third together with the fourth is to the excess of the third above the fourth.
24. The straight lines drawn at right angles to the sides of a triangle from the points of bisection of the sides meet at the same point.
Let ABC be a triangle; bisect BC at D, and bisect CA at E; from D draw a straight line at right angles to BC, and from E draw a straight line at right angles to CA;
let these straight lines meet at G: we have then to shew that the straight line which bisects AB at right angles also passes through G. From the triangles BDG and
